Did you Check your Endpoints?

1st & 2nd Derivative TestsToday you will apply and

formally define the first and second derivative tests.

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Discussion

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1st Derivative Test

Let be a critical number of a function, that is continuous on an open interval containing . If is differentiable on the interval, except possibly at , then( ) can be classified as follows:

1.

c fI c f cf c

If '( ) changes from negative to positive at , then ( ) is a min of .2. If '( ) changes from positive to negative at , then ( ) is a max of .3. If '( ) does not cha

f x c f c relative imum ff x c f c relative imum ff x nge sign at , then ( ) is neither a relative minimum nor a

relative maximum.c f c

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2nd Derivative TestLet be a function such that '( ) 0 and the second derivative of exists on an open intervalcontaining .

1. If "( ) 0, then ( ) is a relative minimum.2. If "( ) 0, then ( ) is a relative maximu

f f cf

c

f c f cf c f c

m.

If "( ) 0, the test fails. In such cases, you can use the 1st derivative test. f c

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Solving Extrema Problems• Identify the quantity to maximize or minimize

and formulate a function in terms of one variable.

• Set the derivative equal to 0 to find the critical numbers. Be sure to look for x-values where the derivative does not exist since critical numbers may occur at those places.

• Use either the first or second derivative test for extrema and be sure to check the endpoints.

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Assignment

HW Q

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