The value of C which satisfies Rolle's Theorem for f(x) = ^4 x + ^4 x in [0, ()/2]. Then C is : Option 1: ()/5 Option 2: ()/3 Option 3: ()/4 Option 4: ()/6

Answer: ()/4. Solution: f'(x) = 4^3 x x - 4^3 x x = -(4x). Set f'(c) = 0: (4c) = 0 4c = n. In (0, /2), c = ()/4.

Calculus

Application of Derivatives

Medium

core

The value of C which satisfies Rolle's Theorem for $f(x) = \sin^4 x + \cos^4 x$ in $\left[0, \frac{\pi}{2}\right]$ . Then C is :

$\frac{\pi}{5}$

$\frac{\pi}{3}$

$\frac{\pi}{4}$

$\frac{\pi}{6}$

✅ Correct Option: 3

$f'(x) = 4\sin^3 x \cos x - 4\cos^3 x \sin x = -\sin(4x)$ .

Set $f'(c) = 0$ : $\sin(4c) = 0 \Rightarrow 4c = n\pi$ .

In $(0, \pi/2)$ , $c = \frac{\pi}{4}$ .

2025: 13 May Shift 2

The absolute maximum value of the function $f(x) = 4x - \frac{1}{2}x^2$ in the interval $\left[-2, \frac{9}{2}\right]$ is

2023: 22 May Shift 3

The maximum value of $2x^3 - 24x + 107$ in the interval $[1, 3]$ is :

2025: 30 May Shift 1

The largest open interval, in which the function $f(x) = \frac{x}{x^2 + 1}$ increases, is

2025: 14 May Shift 2

The function $f(x) = \frac{x - 2}{x + 1}, x \neq -1$ is increasing when (Where $\mathbb{R}$ is a set of real numbers)

2026: 30 May Shift 2

A closed right circular cylinder has a volume of 2156 cm³. The radius of its base so that its total surface area may be minimum, is equal to:

2026: 21 May Shift 1

Match List-I with List-II

List-I	List-II
(A) The interval in which $f(x) = x^2 - 4x + 17$ is increasing.	(I) $(-5,\infty)$
(B) The interval in which $f(x) = x^2 - 4x + 17$ is decreasing.	(II) $(-\infty,-5)$
(C) The interval in which $f(x) = 3x^2 + 30x - 7$ is increasing.	(III) $(2,\infty)$
(D) The interval in which $f(x) = 3x^2 + 30x - 7$ is decreasing.	(IV) $(-\infty,2)$

Choose the correct answer from the options given below: