The function f(x)=_e( x) increases on which of the following intervals for x(-()/2,()/2) ? Option 1: (-()/2,()/2) Option 2: (0,()/4) Option 3: (0,()/2) Option 4: (-()/2,0)

Answer: (-()/2,0). Solution: f'(x)=(- x)/( x)=- x, which is positive only where x<0. Within (-()/2,()/2) that happens for x(-()/2,0).

Calculus

Application of Derivatives

Easy

core

The function $f(x)=\log_e(\cos x)$ increases on which of the following intervals for $x\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ ?

$\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$

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

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

$\left(-\frac{\pi}{2},0\right)$

✅ Correct Option: 4

$f'(x)=\frac{-\sin x}{\cos x}=-\tan x$ , which is positive only where $\tan x<0$ .

Within $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ that happens for $x\in\left(-\frac{\pi}{2},0\right)$ .

2026: 31 May Shift 2

The rate of change in area of a triangle having sides 10cm and 12cm when the variable angle between them at an angle $60°$ , is:

2025: 30 May Shift 2

The function, $f(x) = x - \frac{1}{x}$ is

2022: 25 May Shift 1

The equation of the tangent, to the curve $y = x^2 - 2x - 3$ which is perpendicular to the line $x + 2y + 3 = 0$ , is

2023: 23 May Shift 3

The volume of a cube is increasing at the rate of 27 cm $^3$ /s. How fast is the surface area increasing when the length of the cube is 12 cm.

2026: 30 May Shift 2

If $f(x) = (x+3)e^{-x}$ , then which of the following statements are True?

(A) Function $f(x)$ is increasing on $(-2, \infty)$ .

(B) Function $f(x)$ is decreasing on $(-2, \infty)$ .

(D) Function $f(x)$ is decreasing on $(2, 3)$ .

2025: 22 May Shift 2

The maximum value of $\left(\frac{1}{x}\right)^x$ for $x > 0$ is