The function f(x) = 3x + 3x is (where R is set of real numbers) Option 1: Increasing on R Option 2: Decreasing on R Option 3: Neither increasing nor decreasing on R Option 4: Strictly decreasing on R

Answer: Increasing on R. Solution: f'(x) = 3 - 3 3x = 3(1 - 3x). Since 3x ≤ 1, f'(x) ≥ 0 for every real x, so the function is increasing on R.

Calculus

Application of Derivatives

Easy

core

The function $f(x) = 3x + \cos 3x$ is (where R is set of real numbers)

Increasing on R

Decreasing on R

Neither increasing nor decreasing on R

Strictly decreasing on R

✅ Correct Option: 1

$f'(x) = 3 - 3\sin 3x = 3(1 - \sin 3x)$ . Since $\sin 3x \le 1$ , $f'(x) \ge 0$ for every real $x$ , so the function is increasing on R.

2025: 22 May Shift 2

If the function $f(x) = 2x^2 - kx + 7$ , is increasing on $[1,2]$ , then $k$ lies in the interval

2025: 29 May Shift 2

Let $f(x) = 4x^3 - 18x^2 + 27x - 5$ , $x \in R$ . Then which of the following statements are TRUE?

(A) $f''(x) = 24x - 36$

(B) f has local maxima at $x = \frac{3}{2}$ but no minima

(D) f has both maxima and minima

Choose the correct answer from the options given below:

2026: 14 May Shift 2

For the function $f(x) = \frac{1}{x^2+2x+2}$

which of the following statements are TRUE?

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

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

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

D. maximum value of $f(x)$ is 1

Choose the correct answer from the options given below:

2026: 29 May Shift 1

Match List-I with List-II

Choose the correct answer from the options given below:

2025: 14 May Shift 1

The function $f(x) = tanx - x$

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