`g(x) = x^2 - 2x - 8` Determine the open intervals on whcih 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 does not exist.
A critical value will exist for the first derivative.
If g'(x)>0, the function is increasing in the interval.
If g'(x)<0, the function is decreasing in the interval.
Choose a value for x that is less than 1.
g'(0)=-2 Since g'(0)<0 the function is decreasing in the (-oo,1).
Choose a value for x that is greater than 1.
g'(2)=2 Since g'(2)>0 the function is increasing in the interval (1,`oo).`
Because the function changed direction from decreasing to increasing there exists relative minimum at x=1 and the function is concave up in the interval