`f(x) = -x^3 + 6x^2 - 5` Find the points of inflection and discuss the concavity of the graph of the function.
Find the critical values for x by setting the second derivative of the function equal to zero and solving for the x value(s).
The critical value for the second derivative is x=2.
If f''(x)>0, the curve is concave up in the interval.
If f''(x)<0, the curve is concave down in the interval.
Choose a value for x that is less than 2.
f''(0)=12 Since f''(0)>0 the graph is concave up in the interval (`-oo` 2).
Choose a value for x that is greater than 2.
f''(3)=-6 Since f''(3)<0 the graph is concave down in the interval (2, `oo).`
Since the concavity changed directions and f''(2)=0 the inflection point occurs at x=2. The inflection point is (2, 11).