Question

If f: [-5,5]→ R is a differentiable  function and if f '(x) does not vanish anywhere,prove that

f (-5) ≠ f (5) then

Answer 1

It is given that f : [-5,5] → R is a differentiable function.
Since every differentiable function is a continuous function, we obtain
(a) f is continuous on [−5, 5].
(b) f is differentiable on (−5, 5).
Therefore, by the Mean Value Theorem, there exists c ∈(−5, 5) such that

It is also given that f'( x) does not vanish anywhere.

Hence, proved.