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
When f '(a) = 0, f(x) 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 x = a.
Use the First Derivative Test to investigate the behavior of
, , and at the origin.
Directions: Evaluate f '(x) at two points, one to the left of x = a and the other to the right of x = a.
☛ If the sign changes from negative to positive, then f(x) changes from decreasing to increasing and f(a) is a local min.
☛ If the sign changes from positive to negative, then f(x) changes from increasing to decreasing and f(a) is a local max.
☛ If the sign does not change, then f(x) is 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 f "(a)
☛ If f "(a) > 0, then f(x) is concave up at x = a, and f(a) is a local min.
☛ If f "(a) < 0, then f(x) is concave down at x = a, and f(a) is a local max.
☛ If f "(a) = 0, then this test is inconclusive.