Let I be an open interval, f : I R, and c l. The function

Let I be an open interval, f : I †’ R, and c ˆŠ l. 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| a) If f has a local maximum at c, prove that

for u > 0 and t b) If f is differentiable at c and has a local maximum at c, prove that f'(c) = 0.
c) Make and prove analogous statements for local minima.
d) Show by example that the converses of the statements in parts b) and 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.

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