Absolute+vs.+Relative+Extrema

=Absolute vs Relative Extrema=

Using the First Derivative Test, you have learned how to find the relative maximums and minimums of a function. Now this knowledge can be applied to find the absolute extrema of a function on a closed interval.

The **Relative Extrema** are the minimum and maximum values of a function in a //small interval// of the curve. The **Absolute Extrema** are the minimum and maximum values of the function over the //entire interval// that the curve is defined. There will be only one Absolute Maximum and minimum, while there there can be numerous relative extrema. You should also note that the absolute extrema are relative extrema in and of themselves. Therefore, you can determine which points are the absolute extrema by finding the relative extrema and the endpoints of the function and comparing their values.

Find all Relative and Absolute Extrema of the graph f(x)=(1/3)x 3 -(5/2)x 2 +6x+1 on the closed interval [1,4].
 * Example One **

Solution: We find the derivative using the Power Rule: f '(x)=x 2 -5x+6=(x-2)(x-3) Set the derivative equal to zero to find critical points: 0=(x-2)(x-3) yielding x=2, 3 Find the second derivative: f "(x)=2x-5. Therefore f "(2)=(-1) and f "(3)=1. This means that the graph is concave down at **x=2**, meaning this point is a **Relative Max**, and concave up and **x=3**, meaning this is a **Relative Min**. Now we will test these point and the endpoints to find the Absolute Extrema: f(1)=4.833 f(2)=5.666 f(3)=5.5 f(4)=6.333. We have an **Absolute Minimum at x=1** and an **Absolute Maximum at x=4**.