Calculus

Application of Derivatives

Medium

common

The interval, on which the function $f(x) = x^2e^{-x}$ is increasing, is equal to

$(-\infty,\infty)$

$(-\infty,2) \cup (2,\infty)$

$(-2,0)$

$(0,2)$

✅ Correct Option: 4

A function is increasing on an interval when the derivative is positive: $f'(x) > 0$

Given $f(x) = x^2e^{-x}$ , apply the Product Rule with $u = x^2$ and $v = e^{-x}$ :

$u' = 2x$

$v' = -e^{-x}$

$f'(x) = 2x \cdot e^{-x} + x^2 \cdot (-e^{-x})$

$f'(x) = 2xe^{-x} - x^2e^{-x}$

$f'(x) = e^{-x}(2x - x^2)$

$f'(x) = e^{-x} \cdot x(2 - x)$

For $f'(x) > 0$ :

$e^{-x} \cdot x(2 - x) > 0$

Since $e^{-x} > 0$ for all $x$ , the inequality reduces to:

$x(2 - x) > 0$

The critical points occur when $x = 0$ or $2 - x = 0$ , giving $x = 0$ and $x = 2$ .

Testing the sign of $x(2-x)$ in each interval:

For $x < 0$ : $x$ is negative, $(2-x)$ is positive, so $x(2-x) < 0$

For $0 < x < 2$ : $x$ is positive, $(2-x)$ is positive, so $x(2-x) > 0$

For $x > 2$ : $x$ is positive, $(2-x)$ is negative, so $x(2-x) < 0$

Therefore $f'(x) > 0$ when $0 < x < 2$ .

The function $f(x) = x^2e^{-x}$ is increasing on the interval $(0, 2)$ .

2026: 22 May Shift 1

An angle $\theta, 0<\theta<\frac{\pi}{2}$ , which increases twice as fast as its sine, is

2022: 17 Aug Shift 2

The slope of the tangent to the curve $y = 3x^2 + 2kx - 5$ at $x=1$ is $9$ . The value of $k$ is :

2026: 29 May Shift 2

For the function $f(x)=2x^3-9x^2+12x-5, x \in [0,3]$ , which of the following statements are correct ?

A. The point of absolute maxima is x=3.

B. The absolute minimum value is -1.

C. The critical points are x=1 and x=2.

D. The absolute maximum value is 3.

Choose the correct answer from the options given below:

2026: 19 May Shift 1

For the function $f(x) = \frac{x^4}{4} - 2x^3 + \frac{11}{2}x^2 - 6x$ , which of the following statements are TRUE?

(A) Critical points of $f(x)$ are $x = 1, 2$ and $3$ .

(B) $f(x)$ is increasing over $(1,2)$ .

(D) $f(x)$ is increasing over $(3,\infty)$ .

Choose the correct answer from the options given below:

2026: 29 May Shift 2

A water tank has the shape of an inverted right circular cone with its axis vertical and vertex lower most. Its semi vertical angle is $\tan^{-1}\left(\frac{2}{3}\right)$ . Water is poured into it at a constant rate 6 cubic metre per hour. The rate at which the level of the water is rising at the instant when the depth of water in the tank is 5 meter is

2024: 16 May Shift 1

For the function $f(x) = 2x^3 - 9x^2 + 12x - 5$ , $x \in [0, 3]$ , match List-I with List-II :

List-I	List-II
A. Absolute maximum value	(I) $3$
B. Absolute minimum value	(II) $0$
C. Point of maxima	(III) $-5$
D. Point of minima	(IV) $4$

The interval, on which the function f(x)=x2e−xf(x) = x^2e^{-x}f(x)=x2e−x is increasing, is equal to

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

The interval, on which the function $f(x) = x^2e^{-x}$ is increasing, is equal to