Calculus

Application of Derivatives

Easy

core

Let $f(x) = x^3 - 3ax$ , where $a > 0$ , then the values for which $f(x)$ is decreasing, are

$x < -\sqrt{a}$

$-\sqrt{a} < x < \sqrt{a}$

$x > \sqrt{a}$

$x < -a$

✅ Correct Option: 2

Here $f'(x)=3x^2-3a=3(x^2-a)$ , which is negative precisely when $x^2<a$ . Since $a>0$ , this means $-\sqrt a<x<\sqrt a$ .

2026: 21 May Shift 2

The point on curve $y^2 = 4x$ which is nearest to the point $(2, -8)$ is

2025: 21 May Shift 2

For the function, $f(x) = \frac{-3}{4}x^4 - 8x^3 - \frac{45}{2}x^2 - 350$ , which of the following statements are correct?

(A) $x = -3$ and $x = -5$ are the only critical points of the given function.

(B) $x = -3$ is a point of local minimum.

(D) $x = -5$ is a point of local maximum.

Choose the correct answer from the options given below:

2026: 30 May Shift 1

A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimeters of gas per second. The rate at which the radius of the balloon increases when the radius is 15 cm, is:

2026: 22 May Shift 1

A steel plant is capable of producing 'x' tonnes per day of low-grade steel and 'y' tonnes per day of high-grade steel, where $y=\frac{32-5x}{10-x}$ . If the fixed price of low-grade steel is half that of high-grade steel, then the quantity of low-grade steel that should be produced per day for maximum receipts is

2025: 21 May Shift 2

Match List-I with List-II

List-I (Curve)	List-II (Slope of tangent at $x = 4$ )
(A) $y = \sqrt{x^3}$	(I) -1
(B) $y = \sqrt{x}$	(II) 1
(C) $y = x^3 - 47x$	(III) 1/4
(D) $xy = 16$	(IV) 3

Choose the correct answer from the options given below:

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:

Let f(x)=x3−3axf(x) = x^3 - 3axf(x)=x3−3ax, where a>0a > 0a>0, then the values for which f(x)f(x)f(x) is decreasing, are

Solution

Related questions:

Keyboard Shortcuts

CUET Mathematics Past Year Questions (Topic-Wise)

Let $f(x) = x^3 - 3ax$ , where $a > 0$ , then the values for which $f(x)$ is decreasing, are