This is driving me crazy, because it seems wrong but I haven't found it in any of the errata.

Problem:

Suppose is twice differentiable on 0, and for all . Let be the unique point in at which .

Define

.

Prove that , and that

.

EDIT: Assume

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My issue is that I can only show i.e. I can't get a strict inequality. I would think there is a counterexample if we take so that regardless of which you choose, .