4.4 Concavity and the Second Derivative Test

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Presentation transcript:

4.4 Concavity and the Second Derivative Test Rita Korsunsky

Concavity +

__

A Point of Inflection occurs at the point where the concavity changes. + PI PI PI

Example 1 + _ + Concave down Concave up

Solution: Example 2 f(x) concave up f’’(x)= [f’(x)]’ >0 It means f’(x) is increasing. f’(x) is increasing on (2,4) f(x) is concave up on (2,4) y=f’(x) The graph of f’(x) on the interval [-3,4] is shown above. On what intervals is the graph of f(x) concave up?

2nd Derivative Test for Local Max and Min +

Example 3 - - The critical points are x = -2 , 0 Use second derivative test - - max Therefore, no conclusion Since the 2nd derivative test is not applicable when x = 0, we use the 1st derivative test. - +

For concavity, first find the points of inflection. EXample 3 continues For concavity, first find the points of inflection. 1 - - + Concave down Concave down Concave up +

Constructing first and second derivative graphs Construct f’ if f is given Practice graphing Quiz on derivatives and graphing

4.4 #32 + + + +