Suppose that $x_0$ is stationary pointof a given function $\text{ i.e. } f'(x_0) =0$or in other words, it's differentiable at $x_0$ and$x_0$ is some point where $f$ has a local minimum /maximum, suppose that the second derivative of $f$is continuous in some neighborhood of $x_0$, we have the following :if $f''(x_0) < 0$, then $f$ admits a localmaximum at $x_0$if $f''(x_0) > 0$, then $f$ admits a localminimum at $x_0$My questions : 1 - why do we need the second derivative to be continuous in the neighborhood of $x_0$, what happens if it's not? 2-In general, what kind if informationcan we extract from a continuous derivative ? In particular the last derivative. Thanks in advance for reading (:
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