Find the zeros and the stationary point of the function . Also, find the regions where the function is increasing and decreasing.
You find the zeros where :
Factorize the quadratic polynomial using the quadratic formula or by inspection:
You use the zero product property and find that
You then find that the zeros are , and .
You find the stationary points by solving the equation
You use the quadratic formula and get:
Thus, and .
To find the corresponding -values, you insert the -values into the main function . You get
The stationary points are thus
To determine which type of stationary points they are, you need to find the value of the derivative before, after and between the stationary points. You can do this by drawing a sign chart for the factorized expression . It looks like this:
From the figure, you see that:
decreases on the interval ,
increases on the intervals .
Thus, you know that is a maximum and is a minimum.