Question

The interval in which \(y=x^2e^{-x}\) is increasing is

• A. (-∞,∞)
• B. (−2, 0)
• C. (2,∞)
• D. (0,2)

Answer 1

The Correct Option is D

The correct option is(D): 0,2.

We have,

\( y=x^2e^{-x}\)

∴ \(\frac{dy}{dx}\)\(=2xe^{-x}-x^2e^{-x}=xe^{-x}(2-x)\)

Now, \(\frac{dy}{dx}=0\)

⇒ x=0, and x=2

The points x = 0 and x = 2 divide the real line into three disjoint intervals

i.e.,(-∞,0), (0,2), and(2,∞).

In intervals (-∞,0) and (2,∞), f'(x)<0 as e^-x is always positive.

∴ f is decreasing on (-∞,0) and (2,∞).

In interval (0, 2), f'(x)>0.

f is strictly increasing on (0, 2).

Hence, f is strictly increasing in interval (0, 2). 

The correct answer is D.