Question

The interval in which \( y = x^2 e^x \) is decreasing is _____

• A. \( (-\infty, \infty) \)
• B. \( (2, \infty) \)
• C. \( (-2, 0) \)
• D. \( (0, 2) \)

Hint:

For decreasing intervals, solve \( f'(x)<0 \) using sign chart.

Answer 1

The Correct Option is C

Concept: Function is decreasing when \( \frac{dy}{dx}<0 \)
Step 1: Differentiate. \[ y = x^2 e^x \Rightarrow \frac{dy}{dx} = e^x(x^2 + 2x) \]
Step 2: \[ \frac{dy}{dx} = e^x x(x+2) \]
Step 3: Sign analysis. \[ e^x>0 \Rightarrow x(x+2)<0 \] \[ \Rightarrow -2<x<0 \]