Can a function have a local maximum at a point where $f'(x)$ does not exist?

Yes. Continuous functions do not require differentiability for local extrema. A local maximum exists at $x = c$ if $f(c) \ge f(x)$ for all $x \in (c - \delta, c + \delta)$. For instance, $f(x) = 1 - |x|$ has a sharp corner at $x = 0$ where $f'(0)$ does not exist, yet $x = 0$ is a strict local maximum.

What is the exact geometric difference between a critical point and a point of inflection?

A critical point occurs where $f'(x) = 0$ or $f'(x)$ is undefined, marking potential turning points in functional values. A point of inflection occurs where concavity changes sign—meaning $f''(x)$ changes sign across the point—irrespective of whether $f'(x) = 0$.

How do I solve JEE Advanced calculus inequalities using Application of Derivatives?

Formulate an auxiliary function $g(x) = f(x) - h(x)$ from the target inequality $f(x) > h(x)$. Differentiate to get $g'(x)$ and determine its sign to check monotonicity. Evaluate $g(x)$ at initial boundary anchor points (e.g., $g(0) = 0$). If $g(x)$ is strictly increasing for $x > 0$ and $g(0) = 0$, then $g(x) > 0 \implies f(x) > h(x)$ for all $x > 0$.

Application of Derivatives JEE Notes, Formulas, Traps & PYQs | ExamiFly

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Key Formulas & High-Yield Equations

Equations of Tangent and Normal

$y - y_1 = \left.\frac{dy}{dx}\right|_{P} (x - x_1), \quad y - y_1 = -\frac{1}{\left.\frac{dy}{dx}\right|_{P}} (x - x_1)$

Valid strictly where

f(x)

is differentiable at

P(x_1, y_1)

and

\left.\frac{dy}{dx}\right|_{P} \neq 0

for normal line calculation.

Rolle's Theorem

$f'(c) = 0 \quad \text{for at least one } c \in (a, b)$

Requires

f(x)

to be continuous on

[a, b]

, differentiable on

(a, b)

, and

f(a) = f(b)

at boundary endpoints.

Lagrange's Mean Value Theorem (LMVT)

$f'(c) = \frac{f(b) - f(a)}{b - a} \quad \text{for at least one } c \in (a, b)$

Valid for continuous

f(x)

[a, b]

and differentiable on

(a, b)

. Fails if differentiability breaks at even a single interior point.

Monotonicity Criterion

$f'(x) \ge 0 \implies f(x) \text{ is Increasing on } I$

For strict monotonicity,

f'(x) = 0

must not hold identically over any sub-interval inside domain

I

First Derivative Test for Local Extrema

$f'(x) \text{ changes sign from } + \text{ to } - \text{ across } x = c \implies \text{Local Maximum}$

Applicable at critical points where

f'(c) = 0

or where

f'(c)

is undefined, provided

f(x)

is continuous at

x = c

Global Maximum / Minimum on $[a, b]$

$\text{Global Max} = \max\{f(a), f(b), f(c_1), f(c_2), \dots\}$

Valid for continuous functions on closed bounded domain

[a, b]

, where

c_i \in (a, b)

are internal critical points.

Solved High-Yield Sample Questions

Practice Mode

Question #1JEE Main 2024

Find the maximum area of a rectangle inscribed inside the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with sides parallel to the coordinate axes.

Solution Locked

Step-by-Step Solution & Concept

Step 1

Parametrize the rectangle's top-right vertex on the ellipse as $(a \cos \theta, b \sin \theta)$ for $\theta \in \left(0, \frac{\pi}{2}\right)$ . The length and width of the inscribed rectangle are $2a \cos \theta$ and $2b \sin \theta$ respectively.

Step 2

Express the area function $A(\theta) = (2a \cos \theta)(2b \sin \theta) = 2ab (2 \sin \theta \cos \theta) = 2ab \sin(2\theta)$ .

Step 3

Maximize $A(\theta)$ by taking $\sin(2\theta) = 1$ , which gives the maximum possible area $A_{\max} = 2ab$ when $\theta = \frac{\pi}{4}$ .

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Question #2JEE Main 2023

The function $f(x) = 2x^3 - 9x^2 + 12x + 5$ is strictly decreasing in the interval:

Solution Locked

Step-by-Step Solution & Concept

Step 1

Compute the first derivative $f'(x) = \frac{d}{dx}(2x^3 - 9x^2 + 12x + 5) = 6x^2 - 18x + 12$ .

Step 2

Factorize $f'(x)$ to locate critical points: $f'(x) = 6(x^2 - 3x + 2) = 6(x - 1)(x - 2)$ .

Step 3

For strict decrease, set $f'(x) < 0 \implies 6(x - 1)(x - 2) < 0$ . Solving this inequality yields $x \in (1, 2)$ .

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Mastering Application of Derivatives: Formulas, Traps & Exam Strategy | ExamiFly

Chapter Overview

Core Subtopics & Syllabus Breakdown

Key Formulas & High-Yield Equations

Equations of Tangent and Normal

Rolle's Theorem

Lagrange's Mean Value Theorem (LMVT)

Monotonicity Criterion

First Derivative Test for Local Extrema

Global Maximum / Minimum on $[a, b]$

Common Mistakes & Traps to Avoid

Past Year Question (PYQ) Trends

Solved High-Yield Sample Questions

Frequently Asked Questions

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