1st and 2nd derivative test of max-min

Question

Is there any situation where 1st derivative test is preferred over 2nd derivative test or 2nd derivative test preferred over 1st derivative test? Or do they prefer equivalently?

I have been running across so many problem where the two test can simultaneously used.So I am wondering if there is any situation where I have to choose between the two carefully.

Answer 1

Note that in general, when derivatives exist:

if $f'(x_0)=0$ and $\exists k \geq 2$ s.t. $f^k(x_0) \neq 0$ then

• when the least of those k is even you have a max/min in $x_0$ (depending on the sign)

• when the least of those k is odd you have an inflection point in $x_0$