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Question
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 that f : [-5 , 5 ] → R is 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
`f' (c) = (f(5)- f (-5))/(5 - (-5))`
⇒ 10 f' (c) = f (5) - f (- 5)
It is also given that f'(x) does not vanish anywhere.
∴ f' (c) ≠ 0
⇒ 10 f' (c) ≠ 0
⇒ f (5) - f (-5) ≠ 0
⇒ f (5) ≠ f (-5)
Hence proved.
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