Let $f(x)$ be a continuous function, whose first and second derivatives are continuous on $[0,2π]$ and $f''(x)≥0\:\:∀\:x∈[0,2π]$. Show that$$ \int _0^{2\pi} f(x) \cos x dx \ge 0$$


1 Answer 1


Using integration by parts (twice) we get:

$$\int_{0}^{2\pi} f(x) \cos x \text{ dx} = f'(2\pi) - f'(0) - \int_{0}^{2\pi} f''(x) \cos x \text{ dx}$$

Now $$\int_{0}^{2\pi} f''(x) \cos x \text{ dx} \le \int_{0}^{2\pi} |f''(x) \cos x| \text{ dx} \le \int_{0}^{2\pi} |f''(x)| \text{ dx}$$ $$ = \int_{0}^{2\pi} f''(x) \text{ dx} = f'(2\pi) - f'(0)$$


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