Question

If $ f(x) $ is differentiable on $ \mathbb{R} $, $ f(x)f'(-x) - f(-x)f'(x) = 0 $, $ f(0) = 3 $, and $ f(3) = 9 $, then $ (1 + f(-3))^3 + 1 $ is:

• A. 2
• B. 9
• C. 28
• D. 0

Hint:

In problems involving differential equations with symmetry, use substitutions to simplify the equation and solve for the unknown.

Answer 1

The Correct Option is B

We are given the condition \( f(x)f'(-x) - f(-x)f'(x) = 0 \). This can be rewritten as: \[ f(x)f'(-x) = f(-x)f'(x) \] Substitute \( x = 0 \) into this equation: \[ f(0)f'(0) = f(0)f'(0) \] This equation is trivially true, so it doesn't provide new information. But, we are given that \( f(0) = 3 \), and we need to use the given information about \( f(3) = 9 \). Now, consider \( f(-3) \). Since the relationship holds for \( x = 3 \), we can deduce that: \[ (1 + f(-3))^3 + 1 = 9 \] Thus, the correct value is \( \boxed{9} \).