Question

The value of the real variable \( x > 0 \) that minimizes the function \( f(x) = x^{-x} e^x \) is:

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

Hint:

For optimization problems, take the derivative, set it equal to zero, and solve for critical points. Then, verify whether it minimizes or maximizes the function.

Answer 1

The Correct Option is A

To minimize the function \( f(x) = x^{-x} e^x \), we take the first derivative of the function with respect to \( x \): \[ f'(x) = \frac{d}{dx}\left( x^{-x} e^x \right) \] By applying logarithmic differentiation and simplifying, we find that the critical point occurs at \( x = e \), which minimizes the function. Hence, the value of \( x \) that minimizes the function is \( x = e \).