3.4: Concavity and the Second Derivative Test

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Presentation transcript:

3.4: Concavity and the Second Derivative Test

Objectives Determine intervals on which a function is concave upward or concave downward. Find any points of inflection of the graph of a function.

Concavity Point of Inflection (point where the concavity changes) Concave Up Concave Down

Second Derivative f ’’(x)=f’(f’(x)) Slope of f’ f ’’ is positive if f ’ is increasing (the slope of f is increasing)

Second Derivative f ’’(x)=f’(f’(x)) Slope of f ’ f ’’ is negative if f ’ is decreasing (the slope of f is decreasing)

Theorem 3.7 If f ''(x)>0 for all x in I then the graph of f is concave upward in I. If f ''(x)<0 for all x in I then the graph of f is concave downward in I.

Homework 3.4 (page 189): #1 5-11 odd 15, 17, 21