Bounding techniques

This chapter rigorously introduces fundamental bounding techniques crucial for quantitative analysis in inequalities and estimation. Mastery of upper and lower bounds, range finding, and estimation with parameters is essential for solving complex problems and constitutes a frequently examined component of the CMI BS Hons curriculum.

---

Chapter Contents

| Topic |

|---|-------| | 1 | Upper bounds | | 2 | Lower bounds | | 3 | Range finding | | 4 | Estimation with parameters |

---

We begin with Upper bounds.

Part 1: Upper bounds

Upper Bounds

Overview

Upper-bound questions ask you to show that an expression cannot exceed a certain value. In olympiad-style and exam-style algebra, the real challenge is not just to produce some bound, but to produce a sharp or natural one and justify it cleanly. These problems frequently rely on AM-GM, Cauchy-type thinking, completing the square, or simple monotonicity arguments. ---

Learning Objectives

❗ By the End of This Topic

After studying this topic, you will be able to:

Prove upper bounds using standard inequalities.

Distinguish a valid upper bound from the least upper bound.

Detect sharpness by checking equality cases.

Rewrite expressions into forms where the maximum is visible.

Use domain restrictions correctly in upper-bound problems.

---

Core Idea

📖 Upper Bound

A number $M$ is an upper bound for an expression $E$ on a domain if

$\qquad E \le M$

for every allowed value of the variable.

If equality is possible, then

M

is often the maximum value. ---

Standard Upper-Bound Tools

📐 Method 1: Negative Square Form

If an expression can be written as
$\qquad c-(x-a)^2$
then
$\qquad (x-a)^2\ge 0$
so
$\qquad c-(x-a)^2 \le c$

Hence the maximum value is $c$ .

Example:

\qquad 9-(x-2)^2 \le 9

📐 Method 2: Fixed Sum or Fixed Product

Many upper bounds come from standard inequalities under conditions like:

fixed sum

fixed product

bounded interval

Example:
If

x+y=s

is fixed, then

\qquad xy \le \left(\dfrac{s}{2}\right)^2

This follows from

\qquad (x-y)^2 \ge 0

📐 Method 3: Use the Interval Directly

If $x$ lies in a bounded interval, then expressions like
$\qquad x(1-x),\quad x^2,\quad a-x^2$
can often be bounded directly.

For example, for real $x$ ,
$\qquad x(1-x)=\dfrac14-\left(x-\dfrac12\right)^2 \le \dfrac14$

📐 Method 4: Convert to a Lower-Bound Problem

To prove
$\qquad E \le M$
it is often enough to show
$\qquad M-E \ge 0$

Then use squares or standard inequalities on $M-E$ .

---

Sharpness

❗ Best Possible Upper Bound

If you prove
$\qquad E \le M$
and there exists some allowed value where $E=M$ , then $M$ is sharp.

If equality never occurs, then $M$ may still be an upper bound, but not the maximum.

This distinction matters in high-quality solutions. ---

Minimal Worked Examples

Example 1 Find the maximum value of

\qquad y=5-x^2

Since

x^2\ge 0

\qquad 5-x^2 \le 5

Equality occurs at

x=0

. So the maximum value is

\boxed{5}

. --- Example 2 Find the maximum value of

\qquad y=x(4-x)

Rewrite:

\qquad y=4x-x^2=4-\left(x-2\right)^2

Hence

\qquad y\le 4

Equality occurs at

x=2

. So the maximum is

\boxed{4}

. ---

Upper Bounds from AM-GM

📐 AM-GM for Upper Bounds

AM-GM is often used indirectly.

If $x+y=s$ is fixed and $x,y\ge 0$ , then
$\qquad xy \le \left(\dfrac{s}{2}\right)^2$

Reason:
$\qquad \dfrac{x+y}{2}\ge \sqrt{xy}$

so
$\qquad \dfrac{s}{2}\ge \sqrt{xy}$

hence
$\qquad xy\le \dfrac{s^2}{4}$

This is one of the most important upper-bound facts. ---

Common Patterns

💡 Recognize These Forms

Quadratic opening downward:

\qquad -x^2+bx+c

Product with fixed sum:

\qquad x(s-x)

Expressions on a bounded interval:

\qquad x(1-x),\ 1-x^2,\ (a-x)(x-b)

Symmetric expressions:

reduce with substitutions or identities

---

Common Mistakes

⚠️ Avoid These Errors

❌ Proving an expression is less than something large and calling it the maximum

✅ An upper bound is not always the exact maximum

❌ Forgetting to check equality

✅ Sharpness matters

❌ Using AM-GM without nonnegativity

✅ Check assumptions before applying it

❌ Losing signs while completing the square

✅ Upper-bound problems are very sensitive to sign errors

---

CMI Strategy

💡 How to Attack Upper-Bound Problems

Try to rewrite the expression as a constant minus a square.

If a sum is fixed, try to bound the product.

If the domain is bounded, use it fully.

Check whether the bound is attained.

Separate “upper bound” from “maximum” in your final statement.

---

Practice Questions

:::question type="MCQ" question="The maximum value of

4-x^2

for real

x

is" options=["

2

","

3

","

4

","

5

"] answer="C" hint="Use

x^2\ge 0

." solution="Since

x^2\ge 0

, we have

\qquad 4-x^2 \le 4

Equality occurs at

x=0

. Hence the maximum value is

\boxed{4}

, so the correct option is

\boxed{C}

." ::: :::question type="NAT" question="For real

x

, find the maximum value of

x(2-x)

." answer="1" hint="Complete the square." solution="We write

\qquad x(2-x)=2x-x^2=1-(x-1)^2

Since

(x-1)^2\ge 0

\qquad 1-(x-1)^2 \le 1

Equality occurs at

x=1

. Hence the maximum value is

\boxed{1}

." ::: :::question type="MSQ" question="Which of the following statements are true?" options=["If

E\le M

for all allowed values, then

M

is an upper bound of

E

","If equality occurs in

E\le M

, then

M

is a possible maximum value","For real

x

1-x^2\le 1

","Every upper bound is the least upper bound"] answer="A,B,C" hint="Distinguish upper bound from best upper bound." solution="1. True by definition.

True, because then the bound is attained.

True, since

x^2\ge 0

False, because many upper bounds can exist; only one least upper bound may exist.

Hence the correct answer is

\boxed{A,B,C}

." ::: :::question type="SUB" question="Find the maximum value of

y=3+4x-x^2

for real

x

." answer="

7

" hint="Complete the square." solution="Complete the square:

\qquad y=3+4x-x^2=3-(x^2-4x)=3-\bigl((x-2)^2-4\bigr)

\qquad y=7-(x-2)^2

Since

(x-2)^2\ge 0

\qquad y\le 7

Equality occurs at

x=2

. Therefore the maximum value is

\boxed{7}

." ::: ---

Summary

❗ Key Takeaways for CMI

Upper bounds are often found by rewriting as a constant minus a nonnegative quantity.

Completing the square is one of the cleanest methods.

Fixed-sum problems often lead to product upper bounds.

An upper bound is not automatically the exact maximum.

Equality cases determine sharpness.

---

💡 Next Up

Proceeding to Lower bounds.

---

Part 2: Lower bounds

Lower Bounds

Overview

Lower-bound problems ask for a guaranteed minimum value of an expression. In exam settings, the goal is usually not brute-force optimization but identifying the right inequality, rewriting the expression into a nonnegative form, or using symmetry and equality conditions effectively. CMI-style questions often test whether you can see why a quantity cannot go below a certain value. ---

Learning Objectives

❗ By the End of This Topic

After studying this topic, you will be able to:

Recognize common techniques for proving lower bounds.

Use nonnegative-square arguments to derive bounds.

Apply AM-GM, Cauchy-type ideas, and basic algebraic manipulation carefully.

Track equality cases correctly.

Decide whether a claimed lower bound is sharp.

---

Core Idea

📖 Lower Bound

A number $L$ is called a lower bound of an expression $E$ if

$\qquad E \ge L$

for all values of the variables in the allowed domain.

❗ Best Possible Lower Bound

A lower bound is called sharp or best possible if equality is achieved for some valid choice of variables.

In olympiad-style algebra, finding the lower bound and finding the equality case are both essential parts of the solution. ---

Main Techniques

📐 Technique 1: Nonnegative Squares

The most basic source of lower bounds is

$\qquad (x-y)^2 \ge 0$

From this we get:

$\qquad x^2+y^2 \ge 2xy$

$\qquad (x+y)^2 \ge 4xy$

$\qquad x^2+y^2+z^2 \ge xy+yz+zx$

The last inequality follows from

\qquad (x-y)^2+(y-z)^2+(z-x)^2 \ge 0

📐 Technique 2: AM-GM

For nonnegative real numbers,

$\qquad \dfrac{a+b}{2} \ge \sqrt{ab}$

equivalently,

$\qquad a+b \ge 2\sqrt{ab}$

More generally, for positive numbers,

$\qquad \dfrac{a_1+a_2+\cdots+a_n}{n} \ge \sqrt[n]{a_1a_2\cdots a_n}$

AM-GM is often the fastest way to get a lower bound when the expression is a sum of positive terms. ---

Other High-Value Sources of Lower Bounds

📐 Useful Standard Inequalities

For all real $x$ ,

\qquad x^2 \ge 0

For positive $x$ ,

\qquad x+\dfrac{1}{x} \ge 2

For nonnegative $a,b$ ,

\qquad a^2+b^2 \ge \dfrac{(a+b)^2}{2}

For all real $x,y$ ,

\qquad (x+y)^2 \le 2(x^2+y^2)

For positive $a,b$ ,

\qquad \dfrac{a+b}{2} \ge \sqrt{ab}

---

Equality Cases Matter

❗ Always Ask: When Does Equality Hold?

Typical equality cases:

in $(x-y)^2 \ge 0$ , equality when $\qquad x=y$

in AM-GM for two variables, equality when $\qquad a=b$

\qquad x+\dfrac{1}{x}\ge 2

,
equality when

\qquad x=1

A bound is incomplete if the equality case is ignored.

---

Lower Bound by Rearrangement

💡 Try to Rewrite Around the Suspected Minimum

Suppose you think an expression should be at least $4$ . Then try to write

$\qquad E-4$

as a sum of squares or another nonnegative quantity.

This is one of the most reliable methods in algebraic inequalities.

Example pattern: If

\qquad x^2+4x+7

is suspected to be at least

3

, rewrite:

\qquad x^2+4x+7 = (x+2)^2+3 \ge 3

So the lower bound is

3

, attained at

\qquad x=-2

. ---

Lower Bound by AM-GM

📐 Common AM-GM Patterns

For positive $x$ :

$\qquad x+\dfrac{1}{x} \ge 2$

$\qquad x+y \ge 2\sqrt{xy}$

---

Lower Bounds with Fixed Sum or Fixed Product

❗ Structure Helps

If a sum is fixed, then quadratic expressions often have bounds via square expansion.

Example:
If
$\qquad x+y=s$ ,
then

$\qquad x^2+y^2 = (x+y)^2-2xy = s^2-2xy$

So controlling $xy$ gives a bound.

If a product is fixed, AM-GM or substitution often becomes natural.

---

Minimal Worked Examples

Example 1 Find a lower bound for

\qquad x^2-6x+13

Rewrite:

\qquad x^2-6x+13 = (x-3)^2+4 \ge 4

So the least possible value is

\boxed{4}

. --- Example 2 For positive

x

, find a lower bound of

\qquad x+\dfrac{4}{x}

By AM-GM,

\qquad x+\dfrac{4}{x} \ge 2\sqrt{x\cdot \dfrac{4}{x}} = 2\sqrt{4}=4

So the lower bound is

\boxed{4}

, attained when

\qquad x=\dfrac{4}{x}\implies x=2

---

Common Traps

⚠️ Avoid These Errors

❌ Using AM-GM when variables can be negative

✅ AM-GM needs nonnegative quantities

❌ Giving a lower bound without checking whether it is attained

✅ Equality case is important

❌ Treating a lower bound as the minimum without domain check

✅ Domain matters

❌ Expanding everything when completing the square would be cleaner

✅ Look for structure first

---

CMI Strategy

💡 How to Attack Lower-Bound Problems

Check domain first: are variables real, positive, or nonnegative?

Ask whether completing the square is possible.

Look for AM-GM if there are positive terms or reciprocal structure.

Try to rewrite $E-L$ as a nonnegative expression.

Always state where equality holds.

---

Practice Questions

:::question type="MCQ" question="For real

x

, the least possible value of

x^2+2x+5

is" options=["

3

","

4

","

5

","

2

"] answer="B" hint="Complete the square." solution="We write

\qquad x^2+2x+5=(x+1)^2+4

Since

(x+1)^2\ge 0

, we get

\qquad x^2+2x+5\ge 4

Equality holds at

x=-1

. Hence the correct option is

\boxed{B}

." ::: :::question type="NAT" question="For positive

x

, find the least possible value of

x+\dfrac{1}{x}

." answer="2" hint="Use AM-GM." solution="By AM-GM,

\qquad x+\dfrac{1}{x}\ge 2\sqrt{x\cdot \dfrac{1}{x}}=2

Equality holds when

\qquad x=\dfrac{1}{x}\implies x=1

Hence the least possible value is

\boxed{2}

." ::: :::question type="MSQ" question="Which of the following are always true?" options=["

x^2+y^2\ge 2xy

for all real

x,y

","

x+\dfrac{1}{x}\ge 2

for all positive

x

","

a+b\ge 2\sqrt{ab}

for all nonnegative

a,b

","

x^2-1\ge 0

for all real

x

"] answer="A,B,C" hint="Check the domain and source of each inequality." solution="1. True, from

(x-y)^2\ge 0

True for positive

x

, by AM-GM.

True for nonnegative

a,b

, by AM-GM.

False, for example at

x=0

we get

x^2-1=-1

Hence the correct answer is

\boxed{A,B,C}

." ::: :::question type="SUB" question="Find the least value of

x^2+y^2+2x-4y+10

for real

x,y

." answer="

5

" hint="Complete the square in both variables." solution="Rewrite:

\qquad x^2+y^2+2x-4y+10 = (x^2+2x+1)+(y^2-4y+4)+10-1-4

\qquad = (x+1)^2+(y-2)^2+5

Since both squares are nonnegative,

\qquad (x+1)^2+(y-2)^2+5 \ge 5

Equality holds when

\qquad x=-1,\ y=2

Therefore the least value is

\boxed{5}

." ::: ---

Summary

❗ Key Takeaways for CMI

Lower bounds often come from nonnegative squares or AM-GM.

Rewriting is usually more important than raw calculation.

Equality conditions are part of the answer.

Domain restrictions determine which tools are valid.

A sharp lower bound is one that is actually attained.

---

💡 Next Up

Proceeding to Range finding.

---

Part 3: Range finding

Range Finding

Overview

Range finding is the art of determining all possible values a given expression can take under stated conditions. In exam problems, this is rarely a brute-force computation. The real task is to combine domain restrictions, algebraic rewriting, standard inequalities, and equality cases in a clean way. CMI-style questions often test whether you can convert a messy expression into one whose minimum or maximum is visible. ---

Learning Objectives

❗ By the End of This Topic

After studying this topic, you will be able to:

Identify the domain before attempting to find a range.

Rewrite expressions into forms that reveal lower or upper bounds.

Use squares, AM-GM, and substitutions to find exact ranges.

Track equality cases correctly.

Distinguish between a bound and the actual range.

---

Core Idea

📖 Range

The range of an expression or function is the set of all values it can attain as the variable runs over its domain.

To find a range, you usually need two things:

prove that the expression cannot go below or above certain values

check whether those boundary values are actually attained

---

Step 1: Always Find the Domain

❗ Domain Comes First

Before finding a range, check where the expression is defined.

Typical restrictions:

$\sqrt{A(x)}$ requires $A(x)\ge 0$

A wrong domain almost always gives a wrong range. ---

Main Range-Finding Tools

📐 Method 1: Write as a Square Plus a Constant

If an expression can be written as
$\qquad (x-a)^2 + c$
then
$\qquad (x-a)^2 \ge 0$
so the minimum value is $c$ .

Examples:

$x^2-6x+13 = (x-3)^2+4 \ge 4$

$(x+2)^2-7 \ge -7$

📐 Method 2: Use AM-GM

For positive numbers $u,v$ ,
$\qquad u+v \ge 2\sqrt{uv}$

This is especially useful for expressions like
$\qquad x+\dfrac{1}{x},\quad ax+\dfrac{b}{x},\quad x+y \text{ with } xy \text{ fixed}$

For example, if

x>0

\qquad x+\dfrac{1}{x}\ge 2

with equality at

x=1

📐 Method 3: Use Substitution

If a repeated part appears, set
$\qquad t = g(x)$
and rewrite the expression in terms of $t$ , while also finding the range of $t$ .

Example:
If $y=x^2+x$ , then complete the square:
$\qquad y=\left(x+\dfrac12\right)^2-\dfrac14 \ge -\dfrac14$

📐 Method 4: Use Known Inequalities

Some high-frequency facts:

$(x-a)^2 \ge 0$

$x^2+y^2 \ge 2xy$

for real $x$ , $x^2\ge 0$

for $x>0$ , $x+\dfrac{1}{x}\ge 2$

for real $x$ , $|x|\ge 0$

---

Equality Cases Matter

❗ Do Not Stop at a Bound

If you prove
$\qquad E \ge m$
that only gives a lower bound.

To claim the minimum is exactly $m$ , you must also show that $E=m$ is possible.

The same applies to upper bounds. ---

Minimal Worked Examples

Example 1 Find the range of

\qquad y=x^2-4x+7

Complete the square:

\qquad y=(x-2)^2+3

Since

(x-2)^2\ge 0

, we get

\qquad y\ge 3

Equality occurs at

x=2

. So the range is

\qquad \boxed{[3,\infty)}

--- Example 2 Find the range of

\qquad y=x+\dfrac{1}{x}

for

x>0

. By AM-GM,

\qquad x+\dfrac{1}{x}\ge 2\sqrt{x\cdot \dfrac1x}=2

Equality occurs at

x=1

. As

x\to \infty

y\to\infty

. So the range is

\qquad \boxed{[2,\infty)}

---

Typical Patterns

💡 Recognize These Forms

Quadratic:

\qquad ax^2+bx+c

Reciprocal expressions:

\qquad x+\dfrac1x,\ \ ax+\dfrac{b}{x}

Square root expressions:

analyze domain first, then monotonicity or squaring

Symmetric expressions:

use substitutions or standard inequalities

Rational expressions:

sometimes solve for

x

in terms of

y

and force the discriminant to be nonnegative

---

Discriminant Method

📐 Range via Discriminant

For a rational or algebraic expression, set
$\qquad y=f(x)$
and rearrange to get an equation in $x$ .

Then require that equation to have real solutions, so its discriminant must satisfy
$\qquad \Delta \ge 0$

This often produces the exact range of $y$ .

Example style: If

\qquad y=\dfrac{x+1}{x-1}

, then solve for

x

\qquad y(x-1)=x+1

and then identify the forbidden value of

y

. ---

Common Mistakes

⚠️ Avoid These Errors

❌ Finding a bound without checking whether it is attained

✅ Always check the equality case

❌ Ignoring the domain

✅ Range depends completely on the allowed inputs

❌ Using AM-GM when variables are not positive

✅ AM-GM needs nonnegative or positive quantities

❌ Claiming a minimum for a quadratic without completing the square correctly

✅ Check the algebra carefully

---

CMI Strategy

💡 How to Attack Range Problems

Write the domain first.

Look for a square, AM-GM pattern, or substitution.

Try to reduce the number of moving parts.

Track when equality occurs.

State the final range in interval form whenever possible.

---

Practice Questions

:::question type="MCQ" question="The range of

x^2-6x+10

for real

x

is" options=["

[0,\infty)

","

[1,\infty)

","

[4,\infty)

","

(-\infty,4]

"] answer="B" hint="Complete the square." solution="We write

\qquad x^2-6x+10=(x-3)^2+1

Since

(x-3)^2\ge 0

, the minimum value is

1

, attained at

x=3

. Hence the range is

\boxed{[1,\infty)}

, so the correct option is

\boxed{B}

." ::: :::question type="NAT" question="For

x>0

, find the minimum value of

x+\dfrac{4}{x}

." answer="4" hint="Use AM-GM." solution="By AM-GM,

\qquad x+\dfrac4x \ge 2\sqrt{x\cdot \dfrac4x}=2\sqrt4=4

Equality occurs when

\qquad x=\dfrac4x

\qquad x^2=4

and since

x>0

, we get

x=2

. Hence the minimum value is

\boxed{4}

." ::: :::question type="MSQ" question="Which of the following statements are true?" options=["For real

x

(x-1)^2+5\ge 5

","For

x>0

x+\dfrac1x\ge 2

","The range of

x^2

over all real

x

(0,\infty)

","A lower bound is always the minimum"] answer="A,B" hint="Check equality cases carefully." solution="1. True, because a square is always nonnegative.

True, by AM-GM.

False, because

x^2

can be

0

, so the range is

[0,\infty)

False, because a lower bound need not be attained.

Hence the correct answer is

\boxed{A,B}

." ::: :::question type="SUB" question="Find the range of

y=x^2+2x+5

for real

x

." answer="

[4,\infty)

" hint="Complete the square." solution="Complete the square:

\qquad y=x^2+2x+5=(x+1)^2+4

Since

\qquad (x+1)^2\ge 0

, we get

\qquad y\ge 4

Equality occurs at

\qquad x=-1

Therefore the range is

\qquad \boxed{[4,\infty)}

." ::: ---

Summary

❗ Key Takeaways for CMI

Range finding begins with the domain.

Completing the square and AM-GM are the most common tools.

A bound becomes an extremum only when equality is attainable.

Substitution and discriminant methods are powerful in harder questions.

A clean range argument is usually short but very precise.

---

💡 Next Up

Proceeding to Estimation with parameters.

---

Part 4: Estimation with parameters

Estimation with Parameters

Overview

Estimation with parameters means finding bounds or ranges for expressions that depend on one or more variables together with an extra parameter. The parameter may be fixed, chosen freely, or determined so that an inequality always holds. In exam-level problems, the real challenge is to understand how the parameter changes the strength of the estimate. ---

Learning Objectives

❗ By the End of This Topic

After studying this topic, you will be able to:

Estimate expressions containing parameters by isolating the parameter-dependent part.

Find parameter values for which an inequality holds for all real numbers or all positive numbers.

Use discriminant, AM-GM, and Cauchy-based arguments to handle parameter bounds.

Separate the cases where the parameter is fixed from those where it must be optimized.

Avoid incorrect conclusions caused by ignoring the domain.

---

Core Idea

📖 Parameter-Based Estimation

A parameter is a constant whose value affects the shape or size of an expression.

Typical tasks are:

find the least or greatest value of an expression in terms of a parameter

find all parameter values for which an inequality is true

choose the parameter so that a bound becomes best possible

---

Standard Strategy

💡 Main Plan

When a parameter appears:

decide whether the parameter is fixed or must be found

isolate the variable part of the expression

use a standard inequality or algebraic condition

check equality cases and domain restrictions

---

Estimating Quadratic Expressions with a Parameter

📐 Quadratic Method

If an expression has the form

$\qquad x^2+px+q$

then for a fixed parameter inside $p$ or $q$ , a lower bound can often be found by:

completing the square, or

using the discriminant condition for non-negativity

Key fact: For

\qquad x^2+bx+c\ge 0 \quad \text{for all real }x

it is necessary and sufficient that

\qquad b^2-4c\le 0

when the coefficient of

x^2

1

. ---

Completing the Square with Parameters

📐 Standard Rewrite

For expressions like

$\qquad x^2-2ax+b$

write

$\qquad x^2-2ax+b=(x-a)^2+(b-a^2)$

So the minimum value is

$\qquad b-a^2$

This is one of the most useful school-level estimation methods.

---

Parameter in Rational or Symmetric Bounds

📐 Parameter Matching

If a problem asks for a constant $k$ such that

$\qquad \text{expression}\ge k$

or
$\qquad \text{expression}\le k$

for all allowed values, then:

guess the equality case

test if the expression can be rewritten around that case

choose the parameter to make the inequality sharp

This is common in olympiad-style estimation. ---

Using AM-GM with Parameters

📐 AM-GM Form

For positive variables,

$\qquad u+v\ge 2\sqrt{uv}$

When a parameter is present, choose $u$ and $v$ so that the parameter sits naturally inside one or both terms.

For example, to estimate
$\qquad x+\dfrac{k}{x}$ for $x>0$ ,
AM-GM gives

$\qquad x+\dfrac{k}{x}\ge 2\sqrt{k}$
when $k>0$ .

Equality occurs at

\qquad x=\sqrt{k}

::: ---

Using Cauchy / Titu with Parameters

📐 Typical Form

If
$\qquad \dfrac{x^2}{a+k}+\dfrac{y^2}{b+k}$

appears, and denominators are positive, then think of Titu-type lower bounds:

$\qquad \dfrac{x^2}{a+k}+\dfrac{y^2}{b+k}\ge \dfrac{(x+y)^2}{a+b+2k}$

This is useful when the parameter only shifts denominators. ---

Minimal Worked Examples

Example 1 Find the minimum value of

\qquad x^2-6x+p

for fixed parameter

p

. Complete the square:

\qquad x^2-6x+p=(x-3)^2+(p-9)

Since

(x-3)^2\ge 0

, the minimum value is

\qquad p-9

--- Example 2 Find all real

k

such that

\qquad x^2+2kx+9\ge 0

for all real

x

. For non-negativity of a quadratic for all real

x

, we need the discriminant to be non-positive:

\qquad (2k)^2-4\cdot 1\cdot 9\le 0

\qquad 4k^2-36\le 0

\qquad k^2\le 9

\qquad -3\le k\le 3

Hence the parameter range is

\boxed{[-3,3]}

. ---

Equality Case as a Guide

❗ Best Bound Usually Comes from Equality

When a parameter is to be chosen optimally, equality often indicates the best candidate.

If the guessed equality case is wrong, the parameter choice is usually not sharp.

---

Common Patterns

📐 What Gets Asked Often

find the minimum of an expression in terms of a parameter

find all $k$ such that an inequality holds for all real $x$

estimate $x+\dfrac{k}{x}$ or similar forms

determine the best constant in an inequality

compare several parameter ranges using discriminant or square completion

---

Common Mistakes

⚠️ Avoid These Errors

❌ forgetting whether the parameter is fixed or variable

❌ using AM-GM when positivity is not guaranteed

❌ missing domain restrictions like $x>0$

❌ forgetting to check equality cases

❌ using discriminant incorrectly when the leading coefficient is not positive

---

CMI Strategy

💡 How to Solve Smart

Ask first: am I bounding in $x$ for fixed $k$ , or solving for $k$ ?

For quadratics, complete the square before doing anything else.

For “true for all real $x$ ”, think discriminant.

For expressions like $x+\dfrac{k}{x}$ with $x>0$ , think AM-GM immediately.

For the best constant, look for the equality case and then justify it.

---

Practice Questions

:::question type="MCQ" question="For fixed real parameter

p

, the minimum value of

x^2-4x+p

is" options=["

p-2

","

p-4

","

p+4

","

p

"] answer="B" hint="Complete the square." solution="We write

\qquad x^2-4x+p=(x-2)^2+(p-4)

Since

(x-2)^2\ge 0

, the minimum value is

\qquad p-4

Hence the correct option is

\boxed{B}

." ::: :::question type="NAT" question="Find the least value of

x+\dfrac{9}{x}

for

x>0

." answer="6" hint="Use AM-GM." solution="For

x>0

, by AM-GM,

\qquad x+\dfrac{9}{x}\ge 2\sqrt{x\cdot \dfrac{9}{x}}=2\sqrt{9}=6

Equality occurs when

\qquad x=\dfrac{9}{x}\implies x=3

Therefore the least value is

\boxed{6}

." ::: :::question type="MSQ" question="Which of the following statements are true?" options=["If

x^2+2kx+1\ge 0

for all real

x

, then

|k|\le 1

","For

x>0

x+\dfrac{k}{x}\ge 2\sqrt{k}

whenever

k>0

","The minimum value of

x^2-2ax+b

b-a^2

","AM-GM can be applied to

x+\dfrac{k}{x}

for all real

x

without any sign conditions"] answer="A,B,C" hint="Use discriminant, AM-GM, and square completion." solution="1. True. The discriminant condition gives

\qquad (2k)^2-4\le 0 \implies k^2\le 1

True, provided

x>0

and

k>0

True, since

\qquad x^2-2ax+b=(x-a)^2+(b-a^2)

False. AM-GM requires non-negative quantities, so sign conditions matter.

Hence the correct answer is

\boxed{A,B,C}

." ::: :::question type="SUB" question="Find all real values of

k

such that

x^2+2kx+4\ge 0

for all real

x

." answer="

-2\le k\le 2

" hint="Use the discriminant condition." solution="For

\qquad x^2+2kx+4\ge 0

to hold for all real

x

, the discriminant must satisfy

\qquad (2k)^2-4\cdot 1\cdot 4\le 0

\qquad 4k^2-16\le 0

\qquad k^2\le 4

Hence

\qquad -2\le k\le 2

Therefore the required set of values is

\boxed{[-2,2]}

." ::: ---

Summary

❗ Key Takeaways for CMI

Parameter estimation is about understanding how constants affect bounds.

Completing the square is the fastest tool for quadratic estimation.

Discriminant is essential when an inequality must hold for all real numbers.

AM-GM gives sharp lower bounds in positive-variable parameter problems.

Equality cases usually reveal the best possible parameter or bound.

---

Chapter Summary

❗ Bounding techniques — Key Points

Definition of Bounds: Understand upper and lower bounds, and differentiate them from supremum (least upper bound) and infimum (greatest lower bound). The existence of a bound does not guarantee its attainment.
Techniques for Finding Bounds: Employ calculus (derivatives for extrema), algebraic manipulation (e.g., completing the square, factoring), and fundamental inequalities (e.g., AM-GM, Cauchy-Schwarz, Triangle Inequality, Jensen's Inequality).
Range Finding: Determine the set of all possible output values of a function or expression by systematically identifying its minimum and maximum achievable values within its domain.
Estimation with Parameters: Analyze how bounds are influenced by unknown parameters, often requiring consideration of parameter space, worst-case scenarios, or specific conditions for optimality.
Tightness of Bounds: Emphasize the importance of finding the tightest possible bounds, as these provide the most accurate and useful information for estimation, error analysis, and optimization.
Worst-Case/Best-Case Analysis: Bounding techniques are critical for establishing the extreme limits of a system's behavior, providing guarantees on performance or safety.

---

Chapter Review Questions

:::question type="MCQ" question="Consider the function $f(x) = \frac{x^2+1}{x^2+x+1}$ for $x \in \mathbb{R}$ . What is its maximum value?" options=[" $\frac{1}{3}$ ", " $\frac{2}{3}$ ", " $1$ ", " $2$ "] answer=" $2$ " hint="Rearrange the expression to form a quadratic equation in $x$ and analyze its discriminant for real solutions." solution="Let $y = \frac{x^2+1}{x^2+x+1}$ . Rearranging gives $y(x^2+x+1) = x^2+1$ , which simplifies to $(y-1)x^2 + yx + (y-1) = 0$ . For real values of $x$ , the discriminant must be non-negative: $D = y^2 - 4(y-1)(y-1) \ge 0$ . This simplifies to $y^2 - 4(y-1)^2 \ge 0 \implies y^2 - 4(y^2 - 2y + 1) \ge 0 \implies y^2 - 4y^2 + 8y - 4 \ge 0 \implies -3y^2 + 8y - 4 \ge 0 \implies 3y^2 - 8y + 4 \le 0$ . Factoring the quadratic $3y^2 - 8y + 4 = 0$ yields $(3y-2)(y-2) = 0$ , so $y = \frac{2}{3}$ or $y = 2$ . Since the parabola $3y^2 - 8y + 4$ opens upwards, the inequality $3y^2 - 8y + 4 \le 0$ holds for $\frac{2}{3} \le y \le 2$ . Thus, the maximum value of $f(x)$ is $2$ ."
:::

:::question type="NAT" question="Given positive real numbers $x, y, z$ such that $x+y+z=1$ , what is the minimum value of $\frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ ?" answer="9" hint="Consider applying the Cauchy-Schwarz Inequality or AM-HM Inequality." solution="By the Cauchy-Schwarz Inequality (or Titu's Lemma, a direct consequence):

(x+y+z)\left(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right) \ge (\sqrt{x}\cdot\sqrt{\frac{1}{x}} + \sqrt{y}\cdot\sqrt{\frac{1}{y}} + \sqrt{z}\cdot\sqrt{\frac{1}{z}})^2

(x+y+z)\left(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right) \ge (1+1+1)^2

(1)\left(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right) \ge 3^2

\frac{1}{x} + \frac{1}{y} + \frac{1}{z} \ge 9

Equality holds when

x=y=z=\frac{1}{3}

. Thus, the minimum value is

9

."
:::

:::question type="MCQ" question="Consider the function $g(x) = \frac{x^2+k}{x^2+1}$ for a real constant $k$ . If the least upper bound of $g(x)$ is $5$ , what is the value of $k$ ?" options=[" $1$ ", " $3$ ", " $5$ ", " $9$ "] answer=" $5$ " hint="Rewrite $g(x)$ by adding and subtracting $1$ in the numerator, then analyze its behavior as $x^2$ varies." solution="We can rewrite the function as:

g(x) = \frac{x^2+1 + (k-1)}{x^2+1} = 1 + \frac{k-1}{x^2+1}

Let

u = x^2

. Since

x \in \mathbb{R}

u \ge 0

. So

g(x) = 1 + \frac{k-1}{u+1}

.
If

k-1 > 0

: As

u \to 0

(i.e.,

x \to 0

g(x) \to 1 + \frac{k-1}{1} = 1 + k - 1 = k

. As

u \to \infty

g(x) \to 1 + 0 = 1

. In this case, the maximum value (least upper bound) is

k

.
If

k-1 < 0

: As

u \to 0

(i.e.,

x \to 0

g(x) \to 1 + \frac{k-1}{1} = k

. As

u \to \infty

g(x) \to 1 + 0 = 1

. In this case, the maximum value (least upper bound) is

1

.
If

k-1 = 0

g(x) = 1

for all

x

, so the least upper bound is

1

.
Given that the least upper bound of

g(x)

5

, we must have the first case, where

k-1 > 0

, and thus

k=5

."
:::

---

What's Next?

💡 Continue Your CMI Journey

This chapter on bounding techniques provides foundational tools for rigorous analysis. The principles learned here are indispensable for Inequalities, where the focus shifts to proving relationships between expressions, often by establishing bounds. Furthermore, these techniques are directly applied in Estimation to quantify uncertainty, determine error margins, and approximate complex values, forming the bedrock for robust problem-solving in mathematical finance.

Bounding techniques

Bounding techniques

Chapter Contents

| Topic |

Part 1: Upper bounds

Upper Bounds

Overview

Learning Objectives

Core Idea

Standard Upper-Bound Tools

Sharpness

Minimal Worked Examples

Upper Bounds from AM-GM

Common Patterns

Common Mistakes

CMI Strategy

Practice Questions

Summary

Part 2: Lower bounds

Lower Bounds

Overview

Learning Objectives

Core Idea

Main Techniques

Other High-Value Sources of Lower Bounds

Equality Cases Matter

Lower Bound by Rearrangement

Lower Bound by AM-GM

Lower Bounds with Fixed Sum or Fixed Product

Minimal Worked Examples

Common Traps

CMI Strategy

Practice Questions

Summary

Part 3: Range finding

Range Finding

Overview

Learning Objectives

Core Idea

Step 1: Always Find the Domain

Main Range-Finding Tools

Equality Cases Matter

Minimal Worked Examples

Typical Patterns

Discriminant Method

Common Mistakes

CMI Strategy

Practice Questions

Summary

Part 4: Estimation with parameters

Estimation with Parameters

Overview

Learning Objectives

Core Idea

Standard Strategy

Estimating Quadratic Expressions with a Parameter

Completing the Square with Parameters

Parameter in Rational or Symmetric Bounds

Using AM-GM with Parameters

Using Cauchy / Titu with Parameters

Minimal Worked Examples

Equality Case as a Guide

Common Patterns

Common Mistakes

CMI Strategy

Practice Questions

Summary

Chapter Summary

Chapter Review Questions

What's Next?

🎯 Key Points to Remember

Related Topics in Inequalities and Estimation

Inequality in functions

Standard inequalities

Inequality in sequences and sums

More Resources

Study Notes

Short Notes

Test Series

Mock Tests