Analysis I | Part IA, 2001

Starting from the theorem that any continuous function on a closed and bounded interval attains a maximum value, prove Rolle's Theorem. Deduce the Mean Value Theorem.

Let f:RRf: \mathbb{R} \rightarrow \mathbb{R} be a differentiable function. If f(t)>0f^{\prime}(t)>0 for all tt show that ff is a strictly increasing function.

Conversely, if ff is strictly increasing, is f(t)>0f^{\prime}(t)>0 for all tt ?

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