# 2015-21 Differentiable function

Assume that a function $$f : (0, 1) \to [0, \infty)$$ satisfies $$f(x) = 0$$ at all but countably many points $$x_1, x_2, \cdots$$. Let $$y_n = f(x_n)$$. Prove that, if $$\sum_{n=1}^{\infty} y_n < \infty$$, then $$f$$ is differentiable at some point.

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# 2015-14 Local and absolute maximum

Find all positive integers $$n$$ such that the following statement holds:

Let $$f:\mathbb{R}^n\to \mathbb {R}$$ be a differentiable function that has a unique critical point $$c$$. If $$f$$ has a local maximum at $$c$$, then $$f(c)$$ is an absolute maximum of $$f$$.

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