First Derivative Test for Local Extrema (College Board AP® Calculus AB)

First derivative test

How is the first derivative related to local extrema?

• We know that local extrema (minimums and maximums) are critical points

  • This means the first derivative is equal to zero at these points

• However, there are some points which have a first derivative of zero but are not local extrema

  • E.g. On the graph of , the first derivative is zero at ,

  • but it is not a minimum or maximum, it is a point of inflection

• This means a better definition for local minimums and maximums is:

• If is a critical point of (i.e. ) and if

  • changes sign from positive to negative at ,

    • then has a local maximum at

  • changes sign from negative to positive at ,

    • then has a local minimum at

What is the first derivative test?

• Using the relationship between local extrema and the first derivative described above, we can find local minimums and maximums

• First find the critical points where

• Then find the values of the first derivative:

  • at an value slightly to the left of the critical point

  • at an value slightly to the right of the critical point

• If the first derivative changes (from left to right):

  • from positive to negative, it is a local maximum

  • from negative to positive, it is a local minimum

• If the sign stays the same on both sides of the critical point, it is a point of inflection