## Convex function & limit

Suppose is convex and differentiable with . Prove that .

**Solution:** Since is convex, its derivative is increasing, therefore, .

By the definition of the limit we have that for any there exists such that for any we have there exists $x_0>0$ and such that for any we have . Therefore is strictly increasing, but is bounded above by 0 and large enough. This means that , which means that . This contradicts . The case is solved in a similar way

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Categories: Analysis, Problem Solving
convex, limit

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