Find the maximum area of a parallelogram drawn in the area enclosed by the curves y=4-x^2 & y=x^2+2x

We will use geogebra! Let's see if we can do this.

1) Type in f(x) = 4 - x^2

2) Type in g(x) = x^2 + 2x

3) Use the intersection tool on the two functions and the points A and B will be created.

4) Relabel B to C.

5) Create a slider called b set the interval to -2 to 1 with a step size .001. Type (b,f(b)). Point B will be created on f(x).

6) Use the line tool to create a line from A to B.

7) Use the parallel line tool to create a line that is parallel to AB and passes through C.

8) Get the intersection of this second line and g(x) using the intersection tool. Point D and E will be created. Hide E.

9) Create a line through BC.

10) Draw a line through D that is parallel to BC.

Notice that we now have a generic parallelogram drawn between the two curves.This is all we need!

11) You can hide the lines as best as you can. Create a polygon that uses A,B,C and D as its vertices.

12) Use the slider to get the maximum area. It is not difficult to get 6.75

13) You should have something close to the drawing shown below.

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