Question

If \( P e^x = Q e^{-x} \) for all real values of \( x \), which one of the following statements is true?

• A. \( P = Q = 0 \)
• B. \( P = Q = 1 \)
• C. \( P = 1; Q = -1 \)
• D. \( \frac{P}{Q} = 0 \)

Hint:

When solving functional equations, try substituting specific values of \( x \) (e.g., \( x = 0 \)) and analyze the implications for all values of \( x \).

Answer 1

The Correct Option is A

The given equation is: \[ P e^x = Q e^{-x} \] Rearranging the equation: \[ P e^x - Q e^{-x} = 0 \] For this to hold for all real values of \( x \), both terms must independently be equal to zero. This means: \[ P = 0, \quad Q = 0 \] Thus, the correct answer is \( \mathbf{(A) P = Q = 0} \).