NDA Maths · Application of Derivatives

Monotonicity, Maxima & Minima

The sign of f′ says where a function rises or falls; the zeros of f′ are the candidates for peaks and valleys, sorted by the first- or second-derivative test, with endpoints checked for the absolute extremum.

All Application of Derivatives notes NDA Maths strategy

Why this matters

This is the densest subtopic in the chapter. Almost every question is one of four moves: read intervals from the sign of f′, classify a critical point, find the greatest/least value on an interval, or impose a condition (no extremum / monotonic) on a parameter.

Concept 1 of 4

Increasing and decreasing intervals

Intuition

Where the tangent slopes up the function rises, where it slopes down it falls. So the sign of

f'(x)

on an interval decides monotonicity — find where

f'=0

or is undefined, then test the sign of

f'

in each resulting interval.

Definition

On an interval: $f'(x)>0\Rightarrow f$ increasing; $f'(x)<0\Rightarrow f$ decreasing. Method: solve $f'(x)=0$ for the critical $x$ , split the line at those points, and test the sign of $f'$ in each piece (a product like $(x-a)(x-b)(x-c)$ flips sign at each root). 'Monotonic on an interval' or 'no turning' imposes a one-sided sign condition that may fix a parameter.

Worked example

On which intervals is

f(x)=x^3-3x

increasing?

$f'(x)=3x^2-3=3(x-1)(x+1)$ .
$f'>0$ for $x<-1$ and $x>1$ ; $f'<0$ on $(-1,1)$ .

Answer:Increasing on

(-\infty,-1)

and

(1,\infty)

Practice this conceptself-check · 4 quick reps

Try it yourself

Find where

f(x)=2x^3-9x^2+12x

is increasing.

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
$f'(x)>0$ means $f$ is?
2.
First step to find monotonic intervals?
3.
$f(x)=x^3-3x$ decreasing on?
4.
$x^2-kx$ monotonic increasing on $x>1$ needs?

From the bank · past-year question

Example 1Application of DerivativesMODERATE

f(x) = \dfrac{x^{3}}{3} - \dfrac{5x^{2}}{2} + 6x + 7

increases in the interval

T

and decreases in the interval

S

, then which one of the following is correct ?

[Q100 · Sep · 2019]

Drill 10 more on increasing and decreasing intervals

Concept 2 of 4

Critical points and the derivative tests

Intuition

Local peaks and valleys occur where the tangent is flat (

f'=0

). To tell which is which, either watch the sign of

f'

flip (first-derivative test) or check the bend

f''

(second-derivative test):

f''>0

is a valley,

f''<0

a peak.

Definition

Critical points: where $f'(x)=0$ (or undefined). First-derivative test: $f'$ changes $+\to-$ ⇒ local max; $-\to+$ ⇒ local min. Second-derivative test: at a critical point, $f''>0$ ⇒ local min, $f''<0$ ⇒ local max, $f''=0$ ⇒ inconclusive. A function can attain the same extreme value at two points (e.g. $\pm 3$ ).

Worked example

Find and classify the extrema of

f(x)=x^2+\dfrac{128}{x}

(

x>0

$f'(x)=2x-\dfrac{128}{x^2}=0\Rightarrow x^3=64\Rightarrow x=4$ .
$f''(x)=2+\dfrac{256}{x^3}>0$ , so $x=4$ is a local min; $f(4)=16+32=48$ .

Answer:Local minimum

48

x=4

Practice this conceptself-check · 4 quick reps

Try it yourself

f(x)=x+\dfrac1x

. Classify its critical points.

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Critical points are where?
2.
Second-derivative test: $f''>0$ ⇒?
3.
First-derivative test: $+\to-$ ⇒?
4.
$f''=0$ at a critical point means?

From the bank · past-year question

Example 2Application of DerivativesEASY

A differentiable function

f(x)

has a local maximum at

x=0

. Let

y=2f(x)+ax-b

. Which of the following is/are correct? (A)

f'(0)=0

(B)

f''(0)<0

Select the correct answer using the code given below:

[Q91 · Apr · 2024]

Drill 19 more on critical points and the derivative tests

Concept 3 of 4

Greatest and least value on an interval

Intuition

The absolute (global) maximum or minimum on a closed interval is the largest/smallest among the critical-point values and the endpoint values. Forgetting the endpoints is the classic mistake. On an open interval the extreme may not be attained at all.

Definition

On $[a,b]$ : compute $f$ at every critical point inside, plus $f(a)$ and $f(b)$ ; the greatest is the absolute max, the least the absolute min. On an open interval the sup/inf may be approached but never reached (so 'attains its maximum' can be false even when bounded).

Worked example

Find the greatest and least value of

f(x)=2\sin x+1

[0,\pi]

$f'(x)=2\cos x=0\Rightarrow x=\tfrac\pi2$ ; $f(\tfrac\pi2)=3$ .
Endpoints: $f(0)=1$ , $f(\pi)=1$ .

Answer:Greatest

3

(at

\pi/2

), least

1

(at the endpoints).

Practice this conceptself-check · 4 quick reps

Try it yourself

Does

f(x)=x

attain a maximum on the open interval

(1,2)

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Absolute extremum on $[a,b]$ : check critical points and?
2.
Most common mistake in these problems?
3.
On an open interval, is the sup always attained?
4.
Greatest of $2\sin x+1$ on $[0,\pi]$ ?

From the bank · past-year question

Example 3Application of DerivativesHARD

Let

f(x)=\cos 2x+x

\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]

What is the greatest value of

f(x)

[Q85 · Sep · 2024]

Drill 3 more on greatest and least value on an interval

Concept 4 of 4

Conditions for no extremum / counting extrema

Intuition

To force a polynomial to have no turning points, make its derivative keep one sign — for a cubic, that means the quadratic

f'

has no real roots (discriminant < 0). To count extrema, count the sign-changes of

f'

, i.e. how many times

f'=0

with a genuine sign flip.

Definition

No extremum (cubic): $f'$ is a quadratic; require discriminant $<0$ so $f'$ never changes sign (monotonic).
Counting extrema: solve $f'(x)=0$ on the given domain and count the roots where $f'$ actually changes sign (e.g. $\cos 4x=-\tfrac12$ has several solutions in $(0,\pi)$ ).

Worked example

For what

k

does

f(x)=x^3+x^2+kx

have no local extremum?

$f'(x)=3x^2+2x+k$ . No sign change ⇒ no real roots ⇒ discriminant $<0$ .
$4-12k<0\Rightarrow k>\tfrac13$ .

Answer:

k>\tfrac13

Practice this conceptself-check · 4 quick reps

Try it yourself

How many extreme values does

f(x)=\sin 4x+2x

have on

(0,\pi)

Practice — Level 1 (4 reps)

Quick reps to lock in the method. Try each, then check.

1.
Cubic has no extremum when its $f'$ (a quadratic) has?
2.
Discriminant of $3x^2+2x+k$ ?
3.
To count extrema, count sign-changes of?
4.
$x^3+x^2+kx$ monotonic for $k$ ?

From the bank · past-year question

Example 4Application of DerivativesMODERATE

What is the condition that

f(x)=x^{3}+x^{2}+kx

has no local extremum?

[Q95 · Sep · 2021]

Drill 2 more on conditions for no extremum / counting extrema

Mastery check — 5 interleaved questions

Try each one before clicking. Questions are interleaved across the concepts above, not grouped — interleaving sharpens transfer.

Example 1Application of DerivativesEASY

In which one of the following intervals is the function

f(x)=\frac{x^3}{3}-\frac{7x^2}{2}+6x+5

decreasing?

[Q76 · Apr · 2022]

Example 2Application of DerivativesMODERATE

What is the minimum value of

[x(x-1)+1]^{\frac{1}{3}}

, where

0\leq x\leq 1

[Q94 · Sep · 2018]

Example 3Application of DerivativesEASY

for the items that follow: Let S and T be the sets where

f(x)=\frac{x^3}{3}-\frac{5x^2}{2}+6x+7

decreases and increases respectively.

What is S equal to?

[Q86 · Apr · 2026]

Example 4Application of DerivativesMODERATE

How many extreme values does

\sin4x+2x

, where

0<x<\frac{\pi}{2}

, have?

[Q65 · Apr · 2022]

Example 5Application of DerivativesMODERATE

Consider the following statements: 1.

f(x)=\ln x

is increasing in

(0,\infty)

g(x)=e^{x}+e^{1/x}

is decreasing in

(0,\infty)

Which of the statements given above is/are correct?

[Q91 · Apr · 2023]

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