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?

Practice this conceptself-check · 4 quick reps

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]

Monotonicity is decided by the sign of $f'$ , not $f$

f

increasing

\iff f'(x)\ge 0

on the interval (and

f'(x)\le 0

for decreasing). A large or positive VALUE of

f

says nothing — read the sign of the DERIVATIVE. Sign-test

f'

on each piece between its zeros.

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

Practice this conceptself-check · 4 quick reps

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]

$f''>0$ is a MINIMUM, not a maximum

Second-derivative test: at a critical point

f''>0

means the curve is concave up ⇒ a local minimum;

f''<0

⇒ a local maximum. The sign is the opposite of what students often guess. (

f''=0

is inconclusive — fall back to the first-derivative test.)

$f'=0$ is NECESSARY, not sufficient, for an extremum

A critical point is only a CANDIDATE.

f'=0

can be a point of inflection with no extremum (e.g.

f(x)=x^3

x=0

). You must still confirm a genuine sign change of

f'

(or check

f''

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]

Practice this conceptself-check · 4 quick reps

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]

On a closed interval, ALWAYS test the endpoints

The absolute max/min on

[a,b]

is the largest/smallest among the critical-point values AND

f(a),f(b)

. The extreme value frequently sits at an endpoint, not at a turning point — comparing only critical points is the classic error.

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?

Practice this conceptself-check · 4 quick reps

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]

Count genuine SIGN-CHANGES of $f'$ , not just roots of $f'$

A root of

f'

is an extremum only if

f'

actually flips sign there. A repeated root (e.g.

f'=(x-2)^2

) touches zero without changing sign — no extremum. For a cubic to have NO extremum, force

f'

(a quadratic) to have discriminant

<0

so it never changes sign.

Drill 2 more on conditions for no extremum / counting extrema

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Watch out for (5)

Monotonicity is decided by the sign of $f'$ , not $f$
→ Increasing and decreasing intervals
$f''>0$ is a MINIMUM, not a maximum
→ Critical points and the derivative tests
$f'=0$ is NECESSARY, not sufficient, for an extremum
→ Critical points and the derivative tests
On a closed interval, ALWAYS test the endpoints
→ Greatest and least value on an interval
Count genuine SIGN-CHANGES of $f'$ , not just roots of $f'$
→ Conditions for no extremum / counting extrema

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