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.

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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

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.

Proving an inequality by monotonicity: to show $g(x)>h(x)$ for $x>0$ , set $f=g-h$ ; if $f(0)=0$ (or $f(0)\ge0$ ) and $f'>0$ on $(0,\infty)$ , then $f>0$ there. So $x>\sin x$ for $x>0$ (from $f=x-\sin x$ , $f'=1-\cos x\ge0$ ), $e^x>1+x$ , $\ln(1+x)<x$ . To COMPARE two numbers like $9\ln7$ and $7\ln9$ , find a function of one variable that produces both — $\dfrac{\ln x}{x}$ is decreasing for $x>e$ , so $\dfrac{\ln7}{7}>\dfrac{\ln9}{9}$ , hence $7^9>9^7$ ; equivalently $\dfrac{x}{\ln x}$ is increasing for $x>e$ with minimum $e$ at $x=e$ .
Length of a monotone interval of a periodic function: $\sin kx$ and $\cos kx$ have period $\tfrac{2\pi}{k}$ , so each run of increase or decrease lasts HALF a period, $\tfrac{\pi}{k}$ . Collapse first: $2\cos^2x-1=\cos2x$ decreases on intervals of length $\tfrac\pi2$ , not $\pi$ ; $3\sin x-4\sin^3x=\sin3x$ increases on intervals of length $\tfrac\pi3$ .

Worked example

On which intervals is

f(x)=x^3-3x

increasing?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

NDA · Sep 2019 · Q100Moderate

Example 1 · Application of Derivatives · Monotonicity, Extrema, and Critical Points

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 ?

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 11 more on increasing and decreasing intervals

Concept 2 of 4: Critical points and the derivative tests

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

The same idea in a real exam question:

NDA · Apr 2024 · Q91Easy

Example 2 · Application of Derivatives · Monotonicity, Extrema, and Critical Points

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:

$f''>0$ 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 20 more on critical points and the derivative tests

Concept 3 of 4: Greatest and least value on an interval

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

The same idea in a real exam question:

NDA · Sep 2024 · Q85Hard

Example 3 · Application of Derivatives · Monotonicity, Extrema, and Critical Points

Let

f(x)=\cos 2x+x

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

What is the greatest value of

f(x)

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

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

The same idea in a real exam question:

NDA · Sep 2021 · Q95Moderate

Example 4 · Application of Derivatives · Monotonicity, Extrema, and Critical Points

What is the condition that

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

has no local extremum?

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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