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Let a < b, a, b, ∈ R, and let f : [a, b] → R...

Let a < b, a, b, ∈ R, and let f : [a, b] → R be continuous such that f is twice differentiable on (a, b), meaning f is differentiable on (a, b), and f' is also differentiable on (a, b). Suppose further that there exists c ∈ (a, b) such that f(a) > f(c) and f(c) < f(b).

prove that there exists x ∈ (a, b) such that f'(x)=0.

then prove there exists z ∈ (a, b) such that f''(z)>0.

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