`f(x) = -2x^2 + 4x + 3` Find the critical numbers, open intervals on which the function is increasing or decreasing, apply first derivative test to identify all relative extrema.
Find the critical values for x by setting the first derivative of the function equal to zero and solving for the x value(s).
The critical value for the first derivative is x=1.
If f'(x)>0, the function increases in the interval.
If f'(x)<0, the function decreases in the interval.
Choose a value for x that is less than 1.
f'(0)=4 Since f'(0)>0 the function increases in the interval (-oo, 1).
Choose a value for x that is greater than 1.
f'(2)=-4 Since f'(2)<0 the function decreases in the interval (1, `oo).`
Because the function changed directions from increasing to decreasing a relative maximum will exist. The relative maximum is the point (1, 5).