Assume that exists and (x) > 0 for all x. Show that (x) cannot be negative for all x. Show that '(b) 0

Assume that ƒ" exists and ƒ"(x) > 0 for all x. Show that ƒ(x) cannot be negative for all x. Show that ƒ'(b) ≠ 0 for some b and use the result of Exercise 62 in Section 4.4.

Data From Exercise 62 From Section 4.4

Prove that if ƒ" exists and ƒ"(x) > 0 for all x, then the graph of ƒ“ sits above” its tangent lines.

(a) For any c, set G(x) = f(x) = f'(c)(x - c) - f(c). It is sufficient to prove that G(x) > 0 for all c. Explain why with a sketch. (b) Show that G(c) = G'(c) = 0 and G"(x) > 0 for all x. Conclude that G'(x) < 0 for x < c and G'(x) > 0 for x > c. Then deduce, using the MVT, that G(x) > G(c) for x = c.