Concavity and Second Derivative Test
Definition of Concavity
Let f be differentialbel on an open interval. The graph of f is concave upward if f’ is increasing on the interval and concave downward if f ’ is decreasing.
The definition references when the ____________________________is increasing or decreasing.
How can we tell if the DERIVATIVE is increasing or decreasing?
Test for Concavity
Let f be a function whose second derivative exists on an open interval
If f ‘’(x)>0 then the graph of f is ______________________________________
If f ‘’(x)<0 then the graph of f is ____________________________________________
Points of Inflection
For any value of c that f ‘’(c) = ______, or f ‘’(c) is ______________________ is a point of inflection. This is where the graph will change from concave up to concave down.
Second Derivative Test
You can plug the critical numbers into the second derivative, f ‘’(x).
If f ‘’(c)>0 then that point is a _________________________________________
If f ‘’(c) <0 then that point is a ______________________________________________
Find the points of inflection of f ( x )
Find the relative extrema for f ( x )
3 x 5 +
5 x 3