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The second derivative test is easy to apply once you have found the second derivative, but unfortunately it does not always work. The first derivative test works more often, but is harder to apply. Let's compare the two tests by investigating the graphs of , , and at the origin.

  1. Use the first derivative test to investigate the behavior of , , and at the origin.

  2. Try to confirm your part (a) results 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?

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 .

Directions: Evaluate at two points, one to the left of and the other to the right of .
☛ If the sign changes from negative to positive, then changes from decreasing to increasing and is a local min.
☛ If the sign changes from positive to negative, then changes from increasing to decreasing and is a local max.
☛ If the sign does not change, then is either increasing or decreasing (depending on the sign).

Directions: Evaluate
☛ If , then is concave up at , and is a local min.
☛ If , then is concave down at , and is a local max.
☛ If , then this test is inconclusive.