Calculus

Application of Derivatives

Medium

common

Which of the following functions has a local minima at $x = 0$ ?

(A) $f(x) = x^3$

(B) $f(x) = |x|$

(C) $f(x) = x^2$

(D) $f(x) = x^{-2}$

Choose the correct answer from the options given below:

(A), (B) and (C) only

(D) only

(B), (C) and (D) only

(B) and (C) only

✅ Correct Option: 4

A local minima at $x = 0$ means the function has its lowest point in that neighborhood. Values decrease as they approach $x = 0$ from either side, then increase as they move away.

For $f(x) = x^3$ :

Testing values around zero:

At $x = -1$ : $f(-1) = (-1)^3 = -1$

At $x = 0$ : $f(0) = 0^3 = 0$

At $x = 1$ : $f(1) = (1)^3 = 1$

The function continuously increases from $-1$ to $0$ to $1$ . There is no valley at $x = 0$ .

No local minima at $x = 0$ .

For $f(x) = |x|$ :

Testing values around zero:

At $x = -1$ : $f(-1) = |-1| = 1$

At $x = 0$ : $f(0) = |0| = 0$

At $x = 1$ : $f(1) = |1| = 1$

The function decreases to $0$ , then increases. This creates a V-shape with the lowest point at $x = 0$ .

Has a local minima at $x = 0$ .

For $f(x) = x^2$ :

Testing values around zero:

At $x = -1$ : $f(-1) = (-1)^2 = 1$

At $x = 0$ : $f(0) = 0^2 = 0$

At $x = 1$ : $f(1) = (1)^2 = 1$

The function decreases to $0$ , then increases. This creates a U-shape with the lowest point at $x = 0$ .

Has a local minima at $x = 0$ .

For $f(x) = x^{-2} = \dfrac{1}{x^2}$ :

At $x = 0$ : $f(0) = \dfrac{1}{0^2} = \dfrac{1}{0}$ which is undefined.

The function does not exist at $x = 0$ , so it cannot have a local minima there.

No local minima at $x = 0$ .

Functions with local minima at $x = 0$ : (B) and (C) only

Therefore, the correct answer is the option indicating both (B) and (C).

2023: 30 May Shift 3

The interval in which the function $f(x) = 2x^3 - 3x^2 - 36x + 7$ is strictly decreasing is :

2025: 19 May Shift 1

For the function $f(x) = x^{1/x}$ , $x > 0$ , which of the following are correct?

(A) $x = 0$ is the only point where extremum may occur.

(B) The given function is maximum at $x = e$ .

(D) The maximum value of the function $f(x)$ is $e^{1/e}$ .

Choose the correct answer from the options given below:

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:

2025: 15 May Shift 1

The function $f(x) = x^2 - 4x + 6$ is

(A) Strictly decreasing on $(-\infty, 2) \cup (2, \infty)$

(B) Strictly increasing on $(2, \infty)$

(D) Strictly decreasing on $(-\infty, 2)$

Choose the correct answer from the options given below:

2025: 3 June Shift 2

The two positive numbers whose sum is 16 and the sum of whose squares is minimum then the positive numbers are:

2026: 18 May Shift 1

A solid sphere of gold of radius 5 cm, is being melted in a furnace such that the radius is decreasing uniformly. When the radius is 1cm, the rate at which surface area (A) is decreasing w.r.t. time ( $t$ ) is equal to

( $\frac{dv}{dt}$ = rate at which volume is decreasing w.r.t. $t$ )

Which of the following functions has a local minima at x=0x = 0x=0? (A) f(x)=x3f(x) = x^3f(x)=x3 (B) f(x)=∣x∣f(x) = |x|f(x)=∣x∣ (C) f(x)=x2f(x) = x^2f(x)=x2 (D) f(x)=x−2f(x) = x^{-2}f(x)=x−2 Choose the correct answer from the options given below:

Solution

Related questions:

Keyboard Shortcuts

CUET Mathematics Past Year Questions (Topic-Wise)

Which of the following functions has a local minima at $x = 0$ ?

(A) $f(x) = x^3$

(B) $f(x) = |x|$

(C) $f(x) = x^2$

(D) $f(x) = x^{-2}$

Choose the correct answer from the options given below: