The interval in which the f(x) = x - x, 0 ≤ x ≤ 2 is strictly decreasing is : Option 1: (()/4, (5)/4) Option 2: (0, ()/4) ((5)/4, 2) Option 3: ((3)/4, (7)/4) Option 4: (0, (3)/4) ((7)/4, 2)

Answer: ((3)/4, (7)/4). Solution: f'(x) = x + x = √2(x + ()/4). f'(x) < 0 when (x + /4) < 0, i.e., x + /4 (, 2). So x ((3)/4, (7)/4).

Calculus

Application of Derivatives

Medium

core

The interval in which the $f(x) = \sin x - \cos x$ , $0 \leq x \leq 2\pi$ is strictly decreasing is :

$\left(\frac{\pi}{4}, \frac{5\pi}{4}\right)$

$\left(0, \frac{\pi}{4}\right) \cup \left(\frac{5\pi}{4}, 2\pi\right)$

$\left(\frac{3\pi}{4}, \frac{7\pi}{4}\right)$

$\left(0, \frac{3\pi}{4}\right) \cup \left(\frac{7\pi}{4}, 2\pi\right)$

✅ Correct Option: 3

$f'(x) = \cos x + \sin x = \sqrt{2}\sin\left(x + \frac{\pi}{4}\right)$ .

$f'(x) < 0$ when $\sin(x + \pi/4) < 0$ , i.e., $x + \pi/4 \in (\pi, 2\pi)$ .

So $x \in \left(\frac{3\pi}{4}, \frac{7\pi}{4}\right)$ .

2026: 30 May Shift 2

For which of the following real functions, $x=0$ is a point of local maxima?

(A) $f(x)=4x^4-3x^3-2x^2+1$

(B) $f(x)=3x^5+\sqrt{2}x^4-7x^2+4$

(D) $f(x)=-3x^3+2x^2-9$

Choose the correct answer from the options given below:

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:

2023: 15 June Shift 2

Angle between tangents to the curve $y = x^2 - 5x + 6$ at the points (2, 0) and (3, 0) is :

2025: 13 May Shift 2

The interval in which the function $g(x) = x^2 e^{-x}$ is increasing is:

2022: 17 Aug Shift 2

The slope of the tangent to the curve $y = 3x^2 + 2kx - 5$ at $x=1$ is $9$ . The value of $k$ is :

2026: 26 May Shift 2

Consider the function $f(x) = x^4 - 2x^2$ , then which of the following are TRUE?

A. $f(x)$ is increasing on $(-1,0)$

B. $f(x)$ is decreasing on $(0,1)$

C. $f(x)$ is increasing on $(-\infty,-1)$

D. $f(x)$ is decreasing on $(1,\infty)$

Choose the correct answer from the options given below: