Take a look at the applet: ??
If you have a function with function rule then you can determine the extrema of this function as follows:
find the derivative of the function using differentiation and solve , taking into account the domain of the function;
look at the graph of the derivative function, or set up a sign plot for the derivative;
if moves from negative to positive for (where is within the domain of the function) then has a (local) minimum of ;
if moves from positive to negative at (where is within the domain of the function) then has a (local) maximum of .
If the sign of the derivative does not change, then there are no extrema!
Here you see an example of a sign plot for the derivative.