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 $y=x^2$, $y=x^3$, and $y=x^4$ at the origin.

1. Use the First Derivative Test to investigate the behavior of $y=x^2$, $y=x^3$, and $y=x^4$ at the origin.

   THE FIRST DERIVATIVE TEST

   Directions: Evaluate $f^\prime\left(x\right)$ 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 $f\left(x\right)$ changes from decreasing to increasing and $f\left(a\right)$ is a local min.
   ☛ If the sign changes from positive to negative, then $f\left(x\right)$ changes from increasing to decreasing and $f\left(a\right)$ is a local max.
   ☛ If the sign does not change, then $f\left(x\right)$ is either increasing or decreasing (depending on the sign).

2. 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?

   THE SECOND DERIVATIVE TEST

   Directions: Evaluate $f^{\prime \prime}\left(a\right)$
   ☛ If $f^{\prime \prime}(a)>0$, then $f\left(x\right)$ is concave up at $x = a$, and $f\left(a\right)$ is a local min.
   ☛ If $f^{\prime \prime}(a) < 0$, then $f\left(x\right)$ is concave down at $x = a$, and $f\left(a\right)$ is a local max.
   ☛ If $f^{\prime \prime}\left(a\right) = 0,$ then this test is inconclusive.