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Divergent improper integral

Source: VJIMC 2017, Category I, Problem 4

April 2, 2017
calculusintegrationreal analysiscollege contests

Problem Statement

Let f:(1,∞)→Rf:(1,\infty) \to \mathbb{R} be a continuously differentiable function satisfying f(x)≤x2log⁡(x)f(x) \le x^2 \log(x) and f′(x)>0f'(x)>0 for every x∈(1,∞)x \in (1,\infty). Prove that ∫1∞1f′(x)dx=∞.\int_1^{\infty} \frac{1}{f'(x)} dx=\infty.
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