If f(x) = (x+3)e^(-x), then which of the following statements are True? (A) Function f(x) is increasing on (-2, ∞). (B) Function f(x) is decreasing on (-2, ∞). (C) Function f(x) is increasing on (-∞, -2). (D) Function f(x) is decreasing on (2, 3). Option 1: (A) and (D) only Option 2: (B) and (C) only Option 3: (B), (C) and (D) only Option 4: (A) and (B) only

Answer: (B), (C) and (D) only. Solution: f'(x)=e^(-x)-(x+3)e^(-x)=-(x+2)e^(-x). Since e^(-x)>0, the sign of f' is the sign of -(x+2). So f'>0 for x -2 (decreasing). Hence (B) and (C) hold, and since (2,3)(-2,∞), (D) also holds while (A) fails.

Calculus

Application of Derivatives

Easy

common

If $f(x) = (x+3)e^{-x}$ , then which of the following statements are True?

(A) Function $f(x)$ is increasing on $(-2, \infty)$ .

(B) Function $f(x)$ is decreasing on $(-2, \infty)$ .

(C) Function $f(x)$ is increasing on $(-\infty, -2)$ .

(D) Function $f(x)$ is decreasing on $(2, 3)$ .

(A) and (D) only

(B) and (C) only

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

(A) and (B) only

✅ Correct Option: 3

$f'(x)=e^{-x}-(x+3)e^{-x}=-(x+2)e^{-x}$ . Since $e^{-x}>0$ , the sign of $f'$ is the sign of $-(x+2)$ .

So $f'>0$ for $x<-2$ (increasing) and $f'<0$ for $x>-2$ (decreasing). Hence (B) and (C) hold, and since $(2,3)\subset(-2,\infty)$ , (D) also holds while (A) fails.

2025: 22 May Shift 2

A boat 10 m high floating at a uniform speed of 13 meters per minute(m/min) away from a lamp post 15 m high. Then the rate at which the length of shadow of the boat increases is:

2025: 15 May Shift 2

Match List-I with List-II

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

List-I	List-II
(A) A local maxima at $x =$ ____	(I) $\frac{1}{3}$
(B) A local minima at $x =$ ____	(II) 0
(C) The local minimum value of $f(x) =$ ____	(III) -1
(D) The local maximum value of $f(x) =$ ____	(IV) $-\frac{32}{27}$

Choose the correct answer from the options given below:

2025: 16 May Shift 1

If the interval in which the function f(x) = $\frac{x}{x^2+1}$ is strictly increasing is (-a, a), then a is equal to

2026: 15 May Shift 1

If an open box with a square base is to be made out of a given card board of area $K^2$ square units, then the maximum volume of the box is

2025: 13 May Shift 1

For the function $f(x) = sinx + cosx, x \in [0, \pi]$ , which one of the following is correct?

2025: 21 May Shift 2

For the function $f(x) = 2x^3 - 3x^2 - 12x + 5$ , the difference of maximum and minimum value of $f(x)$ is

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