### Home > APCALC > Chapter 6 > Lesson 6.1.4 > Problem 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.

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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.*

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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.