Finding Maxima and Minima ... The Second Derivative Test

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Let ƒ(x) be a function which is continuous on the interval [-5, 5]. The derivatives of ƒ(x) have the properties indicated in the table below. Draw a sketch of a possible graph of ƒ(x). Assume ƒ(-1) = 0.

First Derivative Test. Suppose that c is a critical point of the function ƒ and suppose that there is an interval (a, b) containing c.

• If ƒ '(x) > 0 for all x in (a, c) and ƒ '(x) < 0 for all x in (c, b), then c is a local maximum of ƒ.

• If ƒ'(x) < 0 for all x in (a, c) and ƒ'(x) > 0 for all x in (c, b), then c is a local minimum of ƒ.

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Can you use the Second Derivative Test to answer these same questions?

Maxima and Minima Practice

First Derivative Test Practice

First Derivative Test Practice

Second Derivative Test. Suppose that c is a critical point of the function ƒ and suppose that there is an interval (a, b) containing c.

• If ƒ'(c) = 0 and ƒ''(c) < 0 then c is a local maximum of ƒ.

• If ƒ'(c) = 0 and ƒ''(c) > 0 then c is a local minimum of ƒ.

The Second Derivative Test

Use the second derivative test to find all local extrema.