Calculus

Application of Derivatives

Medium

common

The function $f(x) = x^2e^{-2x}$ increases on

$(-\infty,\infty)$

$(0, 1)$

$(-\infty, 0) \cup (1, \infty)$

$(1, \infty)$

✅ Correct Option: 2

A function increases where its derivative is positive.

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

Using the product rule with $u = x^2$ and $v = e^{-2x}$ :

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

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

Factoring out $2xe^{-2x}$ :

$f'(x) = 2xe^{-2x}(1 - x)$

For the function to be increasing:

$2xe^{-2x}(1 - x) > 0$

Since $2 > 0$ and $e^{-2x} > 0$ for all real $x$ , the inequality reduces to:

$x(1 - x) > 0$

The product $x(1 - x)$ is positive when both factors have the same sign.

Both positive:

$x > 0$ and $1 - x > 0$

$x > 0$ and $x < 1$

This gives $0 < x < 1$

Both negative:

$x < 0$ and $1 - x < 0$

$x < 0$ and $x > 1$

This is impossible.

The function $f(x) = x^2e^{-2x}$ is increasing on the interval $(0, 1)$ .

2026: 29 May Shift 1

Rate of change of area of a circle with respect to radius

(A) When r = 2 cm is $4\pi$ cm²/cm.

(B) When r = 6 cm is $12\pi$ cm²/cm.

(D) When r = 16 cm is $8\pi$ cm²/cm.

Choose the correct answer from the options given below:

2023: 15 June Shift 2

The value of C which satisfies Rolle's Theorem for $f(x) = \sin^4 x + \cos^4 x$ in $\left[0, \frac{\pi}{2}\right]$ . Then C is :

2025: 3 June Shift 2

Let $f(x) = x^3 - 6x^2 + 9x - 8$ be a function, then which of the following statements are TRUE?

(A) $f'(x) = 3(x - 1)(x - 3)$

(B) The critical points of the function are $x = 1$ and $x = 3$

(D) The local maximum value is $-4$

Choose the correct answer from the options given below:

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:

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 :

2025: 3 June Shift 1

Let f be a function defined by $f(x) = 2x^3 - 3x^2 - 36x + 2$ , then which of the following are correct?

(A) The critical points of f(x) are -2 and 3.

(B) The function f(x) increases in the interval $(3, \infty)$

(D) The function f(x) increases in the interval (-2,3)

Choose the correct answer from the options given below:

The function f(x)=x2e−2xf(x) = x^2e^{-2x}f(x)=x2e−2x increases on

Solution

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The function $f(x) = x^2e^{-2x}$ increases on