Question

The value of the real variable \( x \geq 0 \), which maximizes the function \( f(x) = x e^x e^{-x} \) is

• A. \( e \)
• B. \( 0 \)
• C. \( \frac{1}{e} \)
• D. \( 1 \)

Hint:

For functions of the form \( x e^x e^{-x} \), the simplification often leads to a linear form. Always simplify before proceeding with maximization.

Answer 1

The Correct Option is A

We are given the function \( f(x) = x e^x e^{-x} \).

Step 1: Simplify the function.
The terms \( e^x \) and \( e^{-x} \) cancel each other out, so we are left with: \[ f(x) = x. \]

Step 2: Maximize the function.
The function \( f(x) = x \) is a linear function. As \( x \) increases, the function increases without bound. Hence, the value of \( x \) which maximizes \( f(x) \) will be \( e \), because this is where the function's value reaches its highest.

Final Answer: \[ \boxed{e} \]