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;

Now that we have the function entered into the computer, we can begin work ing to find the maximum, minimum, and inflection points. The first step in finding the max, min, and inflection points is to find the first derivative of the function.

Differentiation

Differentiate the function. To differentiate using Maple we need to use the command "diff( function , variable to differentiate by );" In this case I also named the first derivative dy.

> dy := diff(y, x);

When finding critical points using the first derivative, the next step is to set the derivative to 0 and solve for x. In Maple the way to solve a function and find the x values when the function is equal to zero are very similar. Instead of giving x a value and solving for y, we tell the computer we want y to equal zero. To do this we enter the command "solve({ function }, {x});" Where the x tells Maple to set the function dy equal to zero and solve for x.

> solve({dy}, {x});

Finding the y Component

To find the y component for the ordered pair of maximum and minimum, we substitute x= 2 and x=-1 into the function y.

> subs( x=2, y);

> subs( x=-1, y );

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.

The Second Derivative and Test

Take the second derivative.

> d := diff(dy, x);

Now, let's substitute x=2 and solve for y in the second derivative.

> subs( x=2, d);

This gave us a positive number so we know that when x=2, the function y is at a minimum.

Now, let's substitute x=-1 and solve for y in the second derivative.

> subs( x=-1, d );

This gave us a negative number so we know that when x=-1, the function y is at a maximum.

We now know that the maximum of y is (-1, 12), and the minimum is (2, -15).

The Inflection Point

To find the x point of inflection, we set the second derivative to 0 and solve for x.

> solve({d}, {x});

Last, but not least we substitute x=1/2 into the function y and solve for y.

> subs( x=1/2, y );

We now know that the inflection point of the function y is (1/2, -3/2).

We have completed our task. The maximum, minimum and inflection point of are Max(-1, 12), Min(2, -15), and Inflection(1/2, -3/2).