If is a differentiable function and if does not vanish anywhere, then prove that.

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Solution

It is given that 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 does 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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\int_{2}^{3} f (x) d x = \int_{- 2}^{3}

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