Question

The function \( f(x) = x^2 e^{-x} \), \( x>0 \). Then the maximum value of \( f(x) \) is

• A. 1
• B. \( \frac{1}{e} \)
• C. \( \frac{2}{e} \)
• D. \( \frac{4}{e} \)

Hint:

To find the maximum value of a function, differentiate and set the derivative equal to zero to find the critical points.

Answer 1

The Correct Option is C

Step 1: Finding the critical points.
Differentiate the function \( f(x) = x^2 e^{-x} \) and set the derivative equal to zero to find the critical points. After solving, evaluate the maximum value at the critical point.

Step 2: Conclusion.
Thus, the maximum value of \( f(x) \) is \( \frac{2}{e} \). Hence, the correct answer is option (C).

Final Answer: \[ \boxed{\text{(C) } \frac{2}{e}} \]