Minimizing Area between two functions, one which is constant.

[solomar]

Homework Statement

Graph the curves y = (x+4)(x-5) and y = ax+2 for various values of a. For what value a is the area between the two curves a minimum?

Homework Equations

y = (x+4)(x-5)
y = ax+2

The Attempt at a Solution

For my attempt I attempted to find the primary equation to optimize. I used Area =
$$\int(ax+2-(x+4)(x-5))$$
Since when you graph the two ax+2 is the upper function.
I cannot find a secondary equation to bring it to one variable.

The answer itself is a = -1 (guess and check haha) I am just unsure of how to properly go about minimizing this...
a is what is changing in order for the area to be minimized I understand, and I've exhausted all experimentation with finding a secondary equation in my mind.

Any help or input would be greatly appreciated! This problem is already done on my homework (mymathlab so only the answer is needed), I just really would like to figure out how to properly do this!

 

[Dick]

To solve that problem you have to find the values of x where the two curves intersect in terms of a (use the quadratic equation). Then do the integral between those limits. Finally take the derivative with respect to a and set it equal to zero. It's really quite a mess.

 

[LCKurtz]

To solve that problem you have to find the values of x where the two curves intersect in terms of a (use the quadratic equation). Then do the integral between those limits. Finally take the derivative with respect to a and set it equal to zero. It's really quite a mess.

That's for sure! It's a perfect problem for Maple or its ilk. Maple verifies the OP's intuition that a = -1. Maybe it was a Maple type llab exercise in the first place.