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Curvature and Inflection points Definition If the graph of is lying above its tangent line at every point in an interval , then it is called concave upward (CU) on .

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Definition If the graph of is lying below its tangent line at every point in an interval , then it is called concave downward (CD) on .

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Definition A point where the graph of changes its direction of concavity is called an inflection point.

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Concavity test (a) If for all in an interval ,

then the graph of is CU on . (b) If for all in an interval , then the graph of is CD on .

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EX For each of the following functions, find the intervals where the function is CU, where the function is CD, and the inflection points.

1. 2.

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Second Derivative Test 1. If and , then is a local min. 2. If and , then is a local max.

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EX For the function , find all the local extreme values of

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Curve Sketching

(a) Domain

(b) -interception points and -interception points

(c) Vertical and horizontal asymptotes

(d) Intervals of increase/decrease

(e) Local max and local min

(f) Intervals of concavity and inflection

points.

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I, CU

I, CD

D, CU

D, CD

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EX Sketch the graph of .

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EX Sketch the graph of .

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EX Sketch the graph of

.

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EX Sketch the curve .