Let f (x) be a function on (0, 100), having derivative f (x) and a primitive function F (x) = x f (t)dt 0

defined on the same domain. For all x (0,100), it is known that f(x) 0 and f(x) 0.

(a)We learned that f(x) 0 for all x (0,100) implies that f(x) is decreasing. Prove this statement by using the properties of definite integrals. In other words, for all a,b (0,100), prove that f(a) f(b) if a < b.

(b)We also learned that F(x) = f(x) 0 for all x (0,100) implies that F(x) is concave. Prove this statement by using the properties of definite integrals and results from (a). In other words, for all a, b (0, 100).

Prove that : F ((a+b)/2)1/2(F(a)+F(b))