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

Step 1: Understanding the Concept:
A function is decreasing in the intervals where its first derivative is negative ($\frac{dy}{dx}<0$).
Step 2: Formula Application:
Use the product rule: $\frac{dy}{dx} = x^2 \frac{d}{dx}(e^x) + e^x \frac{d}{dx}(x^2)$ $\frac{dy}{dx} = x^2 e^x + 2x e^x = x e^x (x + 2)$.
Step 3: Explanation:
For decreasing behavior, $x e^x (x + 2)<0$. Since $e^x$ is always positive, we only care about $x(x + 2)<0$. The roots are $x = 0$ and $x = -2$. Using the wavy curve method, the expression is negative between the roots: $-2<x<0$.
Step 4: Final Answer:
The interval is $(-2, 0)$.