Note: In order to broaden our understanding of Maple and critical points, we will be doing the same problem from many different approaches. You may find some to be quicker and "easier" than others. You may also find some to be mo re informative and precise. The way this notebook is set up is to facilitate comparison along these lines. Find what works best for you, and enjoy.

The Function

Enter the function you wish to examine, giving it a name. In this case I called the function "y."

> y := 2*x^3-3*x^2-12*x+5;

Next, you might want to see the plot of the function to get a feel for what the function looks like.

> plot( y(x), x= -3..3 );

Maple's Easy Step:

Finding the y value for the maximum and minimum.

extrema( y, {}, x );

This means that the maximum and minimum occur at y-12 and y=-15.

Now, we have d rawn the function so we know that when y=12 we have a maximum and that when y=-15 we have a minimum. Our next step then is to find the x component for the ordered pair.

Finding the x component.

To find the x component for the ordered pair of maximum and minimum, we substitute y=12 and y=-15 into the function y.

> solve( y=-15, x );

> solve( y=12, x );

Looking at the graph we know that the maximum of y is (-1, 12), and the minimum is (2, -15).

We are now going to pretend we had not graphed y and that we have no way of knowing whether or not y=12 or y=-15 is a maximum or a minimum. Now to find the maximum and minimum we are going to have to take the second derivative of y and use the second derivative test.

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