Extrema are relatively simple. In the graph of a function, there are often a maximum and minimum y or f(x) value. While it is possible to identify the max and min by sight, oftentimes it is not possible.

Theere are also 2 forms of extrema- relative and absolute. Absolute is the max and min for the entire graph, possibly an interval. There can be multiple relatives, but only one absolute. An absolute extrema can also be a relative extrema.

So how do you find the extrema? Well, you need derivatives. Follow the example.

f(x) = x^3 + 8x

f '(x) = 3x^2 + 8

0 = 3x^2 + 8

8 = 3x^2

x = + or - (8/3)^(1/2)

I solved for zero on the derivative to discover where the slope of the equation is zero. My solutions are called critical numbers. Now I must find if they are extrema. Remember those graphs from Algebra II where you put those positive or negative signs? Well, you should. And we're finally going to use them.

.....+..............+......................+

<---------|---------------|-------->

............-(8/3)^(1/2).....(8/3)^(1/2)

Well, there are no absolute or relative extrema for this graph. Why? Well, because the slope's direction never changes. The slope is always increasing, but it "levels off", which is what many x^3 graphs do.

Let's try another:

f(x) = 2x^2 - 3x

f '(x) = 4x - 3

0 = 4x - 3

3 = 4x

x = 4/3

Now, let's test to see if our critical number is an extrema...and whether it is a min or a max.

....-.............+.......

<--------|--------->

............4/3............

Since the slope changes signs, there is an extrema. But is it a min or a max? That example was a minimum....an absolute one. Why? Well, the slope goes from negative to positive. In other words, the parabola was going down on the coordinate plane, hit x = 4/3, and began to move upward. The opposite is true for a max. The slope would go from positive to negative, meaning that the slope was increasing, hit the max, and began to decrease.

But why is it absolute? Well, that's the lowest the function ever goes. If a function has no absolute min or max, we say the limit is infinity...remember limits? I hope you do. Well, that about sums it up for the basics of extrema.