### Home > APCALC > Chapter 5 > Lesson 5.1.4 > Problem 5-42

The Second Derivative Test is easy to apply once you know the second derivative, but unfortunately it does not always work. The First Derivative Test works more often, but is harder to apply. Compare the two tests by investigating the graphs of ,

*, and*

*at the origin.*

When could be at a local max, a local min, or neither (it might be a point of inflection).

The 1st and 2nd derivative tests are ways to tell what is going on at

*.*

Use the First Derivative Test to investigate the behavior of

,, andat the origin.Directions: Evaluate

at two points, one to the left ofand the other to the right of.

☛ If the sign changes from negative to positive, thenchanges from decreasing to increasing andis a local min.

☛ If the sign changes from positive to negative, thenchanges from increasing to decreasing andis a local max.

☛ If the sign does not change, thenis either increasing or decreasing (depending on the sign).Try to confirm your results from part (a) by using the Second Derivative Test. When does the test work? What can you conclude if you know that the second derivative equals zero at the origin?

Directions: Evaluate

☛ If, thenis concave up at, andis a local min.

☛ If, thenis concave down at, andis a local max.

☛ Ifthen this test is inconclusive.