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a) Let f : [a, b] −→ R and g : [a, b] −→ R be...

a) Let f : [a, b] −→ R and g : [a, b] −→ R be differentiable. Then f and g differ by a constant if and only if f ' (x) = g ' (x) for all x ∈ [a, b].

b) For c > 0, prove that the following equation does not have two solutions. x3− 3x + c = 0, 0 < x < 1

c) Let f : [a, b] → R be a differentiable function and let c ∈ (a, b) such that f ' (c) = 0. If f ' (x) > 0 for x < c and f ' (x) < 0 for x > c then show that f has a maximum value at x = c.

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