`f(x) = x^3 - 6x^2 + 12x` 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 down in the interval (-`oo, 2).`
Choose a value for x that is greater that 2.
f''(3)=6 Since f''(3)>0 the graph is concave up in the interval (2, `oo).`