Question

The function $y = xe^{x}$ has}

• A. Minimum value at $x = -1$
• B. Minimum value at $x = 0$
• C. Maximum value at $x = -1$
• D. Maximum value at $x = 0$

Hint:

For $y = x e^{x}$, the critical point is always $x = -1$, and it is always a minimum.

Answer 1

The Correct Option is A

Step 1: Concept
Find the critical points using the first derivative and test for minima/maxima using the second derivative.

Step 2: Meaning
$y' = x e^{x} + e^{x} = e^{x}(x+1)$. Set $y' = 0 \implies x = -1$.

Step 3: Analysis
$y'' = e^{x}(1) + (x+1)e^{x} = e^{x}(x+2)$. At $x = -1$, $y'' = e^{-1}(-1+2) = 1/e$.

Step 4: Conclusion
Since $y'' > 0$ at $x = -1$, the function has a local minimum at $x = -1$. Final Answer: (A)