If is positive for all positive values of and ,

MATHEMATICS · Subjective · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change

Problem

f(x)

is positive for all positive values of

x

and

f'(x) < 0

f''(x) > 0

\forall x \in \mathbb{R}^+

, prove that

\frac{1}{2} f(1) + \int_{1}^{n} f(x)\,dx < \sum_{r=1}^{n} f(r) < \int_{1}^{n} f(x)\,dx + f(1).