(Solution) - a If f x is an even function that is everywhere -(2025 Original AI-Free Solution)

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(a) If f(x) is an even function that is everywhere differentiable, prove that f -(x) is an odd function. Do not assume that f(x) can be expanded in a Taylor series.
(b) Prove that the derivative of an everywhere-differentiable odd function is an even function.
(c) If f(x) is an even function that is differentiable at the origin, find f'(0).