Let I be an open interval, f : I → R, and c ∈ I . The function f is said to have a local m...

Let I be an open interval, f : I → R, and c ∈ I . The function f is said to have a local maximum at c if and only if there is a δ > 0 such that f (c) ≥ f (x) holds for all |x − c| < δ.

a) If f has a local maximum at c, prove that

for u > 0 and t < 0 sufficiently small.

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b) If f is differentiable at c and has a local maximum at c, prove that f ′(c) = 0.

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c) Make and prove analogous statements for local minima.

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d) Show by example that the converses of the statements in parts b) and

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c) are false. Namely, find an f such that f ′(0) = 0 but f has neither a local maximum nor a local minimum at 0.