Question

For any continuous and differentiable function \( f(x) \) with first derivative \( f'(x) \neq 0 \) for all values of \( x \), which one of the following is ALWAYS TRUE for \( a \neq b \)?

• A. \( f(a)>f(b) \)
• B. \( f(a) = f(b) \)
• C. \( f(a) \neq f(b) \)
• D. \( f(a)<f(b) \)

Hint:

If \( f'(x) \neq 0 \) for all \( x \), the function is one-to-one, meaning \( f(a) \neq f(b) \) for \( a \neq b \).

Answer 1

The Correct Option is C

Concept:
If the first derivative \( f'(x) \) is never zero for any value of \( x \), then the function is either strictly increasing or strictly decreasing throughout its domain.

Case 1: Strictly Increasing
If \( f'(x) > 0 \), then for any two values \( a < b \), we have: \[ f(a) < f(b) \] This implies \( f(a) \neq f(b) \).

Case 2: Strictly Decreasing
If \( f'(x) < 0 \), then for any two values \( a < b \), we have: \[ f(a) > f(b) \] Again, \( f(a) \neq f(b) \).

Conclusion:
In either case, the function is injective (one-to-one), meaning it never takes the same value at two different points.

✅ Correct Answer: \( f(a) \neq f(b) \)