Hi tutor, can you please help me to solve that question and show all the work?

Hi tutor, can you please help me to solve that question and show all the work?

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))