(b) Let A € D(RP) and suppose f and g are integrable on A. Let g(x) ≥ 0 for all x € A. If m = inf f(A) and M sup f(A), then there exists a number μ € [m, M] such that S₁ 19 fg = μ 9.

Let f(x)=1/x. Show that f is bounded from above on any interval (α,1), where α>0, but the function is unbounded on (0,1).

Evaluate function given assuming ƒ is integrable on [a, b].

Let ƒ : R → R be a function that is differentiable on (a, ∞), where a is any real constant. Let g : R → R be a function defined by g(x) = f(x+1) – f(x). If limä→∞ ƒ'(x) = 0, prove that limä→∞ g(x) = 0 by using the Mean Value Theo- rem.