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Question
Does there exist a differentiable function f(x) such that f(0) = – 1, f(2) = 4 and f(x) ≤ 2 for all x. Justify your answer
Solution
Given: For f(x), f'(x) ≤ 2, f(0) = –1, f(2) = 4
∴ a = 0, b = 2
By Lagrange’s Mean Value Theorem,
f(b) – f(a) ≤ f'(c)(b – a)
f(2) – f(0) ≤ f'(c)(2 – 0)
`(4 + 1)/2` ≤ f'(c)
⇒ `5/2` ≤ f'(c) ≤ 2 ......(Given)
f(x) cannot be a differentiable function in (0, 2) as f'(x) cannot be 2.5.
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