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.