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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.

demon
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  • In generale you need both of them to understand what kind of point you are facing with. – user Dec 14 '17 at 18:50
  • I prefer 1st derivative test. For some functions, computing the second derivative can be time-consuming, and the task of finding the zeros of the 1st derivative is almost as hard than studying the sign. – ajotatxe Dec 14 '17 at 18:52
  • @ajotatxe yeah. I have been using both the tests according to the form of the function and the critical points... – demon Dec 15 '17 at 19:12

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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 $x_0$

user
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