The function f(x) = 4x + 1/x is (A). Increasing in the interval (-∞, -1/2) (B). Increasing in the interval (-1/2, 0) (C). Decreasing in the interval (0, 1/2) (D). Decreasing in the interval (1/2, ∞) Choose the correct answer from the options given below: Option 1: (B) and (D) only Option 2: (A) and (C) only Option 3: (B) and (C) only Option 4: (C) and (D) only

Answer: (A) and (C) only. Solution: f'(x)=4-1/x^2=(4x^2-1)/x^2. This is positive when, x, >12 and negative when 0<, x, <12. So f increases on (-∞,-12) and (12,∞) and decreases on (-12,0) and (0,12). Hence (A) and (C) are correct.

Calculus

Application of Derivatives

Medium

common

The function $f(x) = 4x + \frac{1}{x}$ is

(A). Increasing in the interval $\left(-\infty, -\frac{1}{2}\right)$

(B). Increasing in the interval $\left(-\frac{1}{2}, 0\right)$

(C). Decreasing in the interval $\left(0, \frac{1}{2}\right)$

(D). Decreasing in the interval $\left(\frac{1}{2}, \infty\right)$

Choose the correct answer from the options given below:

(B) and (D) only

(A) and (C) only

(B) and (C) only

✅ Correct Option: 2

$f'(x)=4-\frac{1}{x^2}=\frac{4x^2-1}{x^2}$ . This is positive when $|x|>\frac12$ and negative when $0<|x|<\frac12$ . So $f$ increases on $(-\infty,-\frac12)$ and $(\frac12,\infty)$ and decreases on $(-\frac12,0)$ and $(0,\frac12)$ . Hence (A) and (C) are correct.

2025: 15 May Shift 1

Which of the following functions are increasing on $x \in \left(0, \frac{\pi}{2}\right)$ ?

(A) $f(x) = \sin x$

(B) $f(x) = \cos x$

(D) $f(x) = \cos 3x$

Choose the correct answer from the options given below:

2026: 19 May Shift 1

The interval in which the function $f(x) = \log_e(2+x) - \frac{2x}{2+x}$ is increasing is

2025: 29 May Shift 2

The function $f(x) = x^2 - x + 1$ is

2025: 14 May Shift 1

The rate of change of area of a circle with respect to its circumference when radius is 4cm, is

2025: 22 May Shift 1

The equation of the tangent line to the curve $y = x^2 - 2x + 5$ which is parallel to the line $4x - y + 1 = 0$ is

2025: 13 May Shift 1

Consider the function $f(x) = x^3 - 3x$ . Then

Match List-I with List-II

List-I	List-II
(A) Point of local Maxima	(I) 1
(B) Point of local Minima	(II) -1
(C) Local maximum value	(III) 2
(D) Local minimum value	(IV) -2

Choose the correct answer from the options given below:

Solution

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