Date: MATH 1210-4 - Spring 2004
Note: Unless stated otherwise, answers without justification receive no credit. Please, show your work!
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 .
We find this by analyzing the sign of . Since vanishes at , it suffices to evaluate
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
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 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 .
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.