CPM Homework Help : CALC Problem 5-42

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

Use the first derivative test to investigate the behavior of $y = x^2$ , $y = x^3$ , and $y = x^4$ at the origin.
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?

Hint:

When $f^{\prime\prime}(a) = 0$ , $f(x)$ 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 $x = a$ .

THE FIRST DERIVATIVE TEST

Directions: Evaluate $f^\prime(x)$ 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).

THE SECOND DERIVATIVE TEST

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