Prove that if exists and (x) > 0 for all x, then the graph of sits above its tangent lines. (a) For any c, set

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.