Calculus

Application of Derivatives

Easy

common

Let $f(x)$ be a function defined on an interval $[a,b]$ such that $f(x)$ is differentiable on $(a,b)$ . Then

Match the LIST-I with LIST-II

LIST-I
Where $I$ is subset of $[a,b]$ LIST-II
A. function $f(x)$ is increasing on interval $I$ I. $f'(x) < 0\ \ \forall x \in I$
B. function $f(x)$ has local maxima at $x = a$ II. $f'(a) = 0\ \&\ f''(a) > 0$
C. function $f(x)$ has local minima at $x = a$ III. $f'(x) > 0\ \ \forall x \in I$
D. function $f(x)$ is decreasing on interval $I$ IV. $f'(a) = 0\ \&\ f''(a) < 0$

Choose the correct answer from the options given below:

	LIST-I Where $I$ is subset of $[a,b]$		LIST-II
A.	function $f(x)$ is increasing on interval $I$	I.	$f'(x) < 0\ \ \forall x \in I$
B.	function $f(x)$ has local maxima at $x = a$	II.	$f'(a) = 0\ \&\ f''(a) > 0$
C.	function $f(x)$ has local minima at $x = a$	III.	$f'(x) > 0\ \ \forall x \in I$
D.	function $f(x)$ is decreasing on interval $I$	IV.	$f'(a) = 0\ \&\ f''(a) < 0$

A-IV, B-III, C-II, D-I

A-I, B-IV, C-II, D-III

A-III, B-II, C-IV, D-I

A-III, B-IV, C-II, D-I

✅ Correct Option: 4

A positive derivative throughout $I$ means increasing and a negative one means decreasing. By the second derivative test, a stationary point with $f''(a)<0$ is a local maximum and with $f''(a)>0$ a local minimum. Hence A-III, B-IV, C-II, D-I.

2022: 17 Aug Shift 2

If the function $f(x) = x^2 - ax - 2$ is strictly decreasing on $(2, 3)$ then $a$ lies in the interval.

2025: 19 May Shift 1

The semi vertical angle of a right circular cone of maximum volume of a given slant height is

2025: 22 May Shift 2

Match List-I with List-II

List-I	List-II
(A) Marginal average cost if cost function $C(x) = \frac{50}{\sqrt{x}}$	(I) $50\sqrt{x}$
(B) Marginal average cost if cost function $C(x) = 50\sqrt{x}$	(II) $-\frac{75}{x^2\sqrt{x}}$
(C) Revenue function if demand function $P=\frac{50}{\sqrt{x}}$	(III) $\frac{-25}{x\sqrt{x}}$
(D) Marginal revenue if demand function $P=50\sqrt{x}$	(IV) $75\sqrt{x}$

Choose the correct answer from the options given below:

2022: 23 Aug Shift 1

Distance between Sumit and Amit in terms of $x$ is :

2025: 22 May Shift 1

Match List-I with List-II

List-I	List-II
Function	Increasing on the interval
(A) $f(x) = -x^2 - 2x + 1$	(I) $(-\infty, -1)$
(B) $f(x) = x^2 + 1$	(II) $(1, \infty)$
(C) $f(x) = x^2 - 2x + 3$	(III) $(-\infty, 0)$
(D) $f(x) = -x^2$	(IV) $(0, \infty)$

Choose the correct answer from the options given below:

2026: 21 May Shift 1

The maximum and the minimum value of the function $f(x) = -||x+3|-5| + 2$ are :

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