`f(x) = -x^3 + 6x^2 - 9x - 1` Determine the open intervals on which the graph is concave upward or downward.
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).`