Show that $\int_1^3f(x)dx+\int_{11}^{13}f(x)dx\ge\int_5^9f(x)dx$
Solution 1:
Let $l$ be the straight line through the points $(5,f(5)), (9,f(9))$.
Note that $\int_1^3 l(x)dx + \int_{11}^{13} l(x)dx = \int_5^9 l(x)dx$.
Since $f$ is convex, $f(x) \le l(x)$ for $x \in [5,9]$ and $f(x) \ge l(x)$ otherwise.