If `f : [-5, 5] to R` is a differentiable function and if `f^(prime)(x)` does not vanish anywhere, then prove that `f(-5) ne f(5)`.

Explanation

To prove that \( f(-5) \neq f(5) \) given that \( f : [-5, 5] \to \mathbb{R} \) is a differentiable function and \( f'(x) \) does not vanish anywhere, we can follow these steps:

### Step 1: Understand the implications of \( f'(x) \neq 0 \)

Since \( f'(x) \) does not vanish anywhere on the interval \([-5, 5]\), it means that the derivative is either always positive or always negative throughout this interval. This indicates that the function \( f(x) \) is either strictly increasing or strictly decreasing.

### Step 2: Assume \( f'(x) > 0 \) (strictly increasing case)

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