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Question

# Suppose that a function f is differentiable at $x_0$ and that $f^{\prime}\left(x_0\right)>0$. Prove that there exists an open interval containing $x_0$ such that if $x_1$ and $x_2$ are any two points in this interval with $x_1, then $f\left(x_1\right).

Solution

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Given that the $f'(x_0)>0$ means that the slope of the tangent line to the graph at that point is positive.

It means that the value of the function as $x$ increases also increases.

$x\text{ increases}=f(x)\text{ increases}$

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