CPM Homework Banner
6-50.

Write the first and second derivatives of each function. Use these derivatives to test for maxima and minima in the given interval. Remember to check the endpoints.

Since this is an OPEN interval, you are checking for LOCAL maxima and minima only.

You can use the 1st- or the 2nd-Derivative Test.
No matter which test you choose, your first step will be to find extrema CANDIDATES by determining where AND where DNE.

1st-Derivative Test: Check values to the left and the right of each candidate.
If changes from positive to negative, then you found a local maximum.
If changes from negative to positive, then you found a local minimum.
If does not change sign, then you found a point of inflection.

2nd-Derivative Test: Evaluate each candidate in the 2nd-derivative.
If is negative, then you found a local maximum.
If is positive, then you found a local minimum.
If  or DNE, then this test is inconclusive.

There is a local minimum at .
But recall that a local min or max is a -value, not an -value.

Since this is a CLOSED interval, you are checking for both LOCAL and GLOBAL maxima and minima.

Find the local maxima and minima. (Refer to the hints in part (a) for guidance.)
Note: These are also global candidates.

Find the global max and min by evaluating the y-value of each candidate.
The highest -value is the global maximum.
The lowest -value is the global minimum.

It is very important to consider candidates where
DNE.
Recall that a derivative will not exist at a cusp, endpoint, jump, hole or vertical tangent.
SO CONSIDER THE ENDPOINTS AS CANDIDATES.