*Date:* MATH 1210-4 - Spring 2004

*Note:* Unless stated otherwise, answers without justification
receive *no credit*. Please, *show your work!*

- Let
- Find all the critical points of .
We have . Then, the stationary points are , but vanishes only if or . Then, the stationary points are .

Since there are no endpoints nor singular points, the only critical points are .

- Find the regions where is increasing and where it is
decreasing.
We find this by analyzing the sign of . Since vanishes at , it suffices to evaluate

to conclude that is increasing on and on , while it is decreasing on . - Find the regions where is concave up and where it is
concave down.
We find the concavity of by studying the sign of . As usual, we start by finding where , that is where vanishes, which only happens if or . Then, it suffices to evaluate

to conclude that is concave up over and while it is concave down over and . - Find all inflection points of .
An inflection point is a point where the concavity of changes, and this happens at , and . These are the three inflection points of .

- Using the first or second derivatives of decide which
of the critical points are local maximum values and which are
local minimum values.
Using the first derivative we see that, coming from the left, changes from increasing to decreasing at , so it is a local maximum; at continues decreasing (so doesn't have an extremum value there), and at goes from decreasing to increasing so that at has a local minimum.

The local maximum value is and the local minimum value is .

Alternatively, the same results for and are obtained from and . But, the second derivative test doesn't decide the behavior of at since .

- Using only the information that you have gathered so far,
sketch a graph of showing this information.
Figure 1 shows the sketch, where the critical and inflection points are marked. Notice that the increasing and decreasing regions, as well as the concavity matches our results from the previous items.

- Find all the critical points of .

Javier Fernandez 2004-03-24