Hi, jackster18.

Do you know what a critical number is? We say that is a critical number of if or if is undefined at . Knowing this, can you see how to find the critical points of a function by looking at the graph of its derivative?

As for maxima and minima, note that a function will have a relative maximum or minimum only at its critical numbers (but not all critical numbers will necessarily have an extremum). The second derivative test says that if is a critical number of , then implies is a relative minimum, implies is a relative maximum, and means the test fails.

So, find the critical numbers from the graph of , and then look at the sign of the second derivative at those points.

If the second derivative test fails, you can use the first derivative test: if is a critical number of and changes from positive to negative or from negative to positive at , then must be a relative extremum.

If, for a function , on an interval, then is increasing on that interval and vice versa.