Do we really need Second Derivative Test?

Question

When drawing a graph of a real function $f$, we first compute $f'$, and solve the equation $f'(x)=0$. For example, we get the solutions are $x=3,9$. Next, we want to decide whether $f$ really attains a relative extremum at $3$ and $9$. We can use the First Derivative Test, and see whether there is a sign change at $3$ and $9$. It seems that the Second Derivative Test is not necessary, but some authors said that sometimes the Second Derivative Test is more applicable than the First Derivative Test. I don't know why it is so. As in my example, is this because sometimes judging whether the sign changes at $3,9$ is hard? Or is it due to other reasons? If you can give some concrete examples, it would be very nice. :)

Answer 1

Here is a more nasty example: $$f(x)=\sin\left(\frac{1}{x}\right),\quad x\neq 0.$$ The derivative is $$f'(x)=-\frac{1}{x^2}\cos\frac{1}{x}.$$ Hence if $f'(x_e)=0$, you have $\cos\frac{1}{x_e}=0$. The second derivative is $$f''(x)=\frac{2}{x^3}\cos\frac{1}{x} - \frac{1}{x^4}\sin\frac{1}{x}.$$ Inserting $x_e$ now gives you $f''(x_e)=-\frac{1}{x_e^4}\sin\frac{1}{x_e}.$

This second deivative test works far more reliably in this case, because you have an infinite number of extremal points which are getting closer, when $x$ tends to zero. This means finding points in a neighbourhood of a possible extremum to test for a sign change is more difficult here (it can be done, but is tedious).

Another more advanced reason for using the second derivative is, that this method can be generalised to higher dimensions. For the sign change method this is not possible. (see e.g.https://en.wikipedia.org/wiki/Second_partial_derivative_test)