GATE 2025 BM – Question 18
A real function f is continuous and differentiable with f′(x)≠0 everywhere. For finite real a≠b, which always holds?
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Correct answer: (C) f(a)≠f(b)
Explanation
If f(a)=f(b) for distinct a,b, Rolle’s theorem on the interval between them would give an interior c with f′(c)=0, contradicting the premise. Thus f(a)≠f(b), C. The direction of the inequality depends on a,b and the derivative sign.