3. Let I be an open interval, f : 1--+ R, and c E I. The function f is said to have a local maximum at c if and only if there is a 8 > 0 such that f

(c) ~ f(x) holds for all Ix - cl < 8.

(a) If f has a local maximum at

c, prove that

.f:..(...c.:. ..+.---: hh:-)-- -.-.:.....f.:(...c..:)... -< 0 and f(c + H) - f

(c) > 0 H -

for h > 0 and H < 0 sufficiently small.

(b) If f is differentiable and has a local maximum at

c, prove that f'

(c) = O.

(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 1'(0) = 0 but f has neither a local maximum nor a local minimum at O.