If f : [−5, 5] → R is differentiable and if f' (x) doesnot vanish anywhere, then prove that f (−5) ± f (5).

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Solution

It is given thatis a differentiable function.
Every differentiable function is a continuous function. Thus,
(a) f is continuous in [−5, 5].
(b) f is differentiable in (−5, 5).
Therefore, by the Mean Value Theorem, there exists c ∈ (−5, 5) such that

It is also given thatdoes not vanish anywhere.

Hence proved.

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If $f : [- 5, 5] \to R$ is a differentiable function and if $f (x)$ does not vanish anywhere, then prove that $\int (- 5) \neq \int (5)$ .

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