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by | Jun 24, 2023 | Posted Assignments

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))

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