I am attempting to use the second derivative test (which gives you concavity of a function) to find the relative extrema (minimums/maximums) on an interval.

I already used the second derivative test to find the concavity of the function, and this is what I got:

(-∞, 1] concave downward
[1, ∞) concave upward

I know that using the first derivative test you can get relative extrema by checking if the slope of each side of the point is increasing or decreasing, but how would I get this with the second derivative test using concavity instead?


1 Answer 1


After finding the extrema using the first derivative test, you can find out what kind of an extrema it is according to the value of the second derivative at that point: If the second derivative is larger than 0, the extrema is a minimum, and if it is smaller than 0(negative), the extrema is a maximum.

  • $\begingroup$ I would find all the critical numbers using f'(x) and then plug them into f''(x), correct? $\endgroup$
    – JohnDoe
    Oct 22, 2015 at 23:44
  • 1
    $\begingroup$ Yes. Find the zeroes of $f'(x)$, insert the values into $f''(x)$ and figure out whether they are a maximum or a minimum according to my answer. $\endgroup$
    – akukas
    Oct 22, 2015 at 23:48

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