# Maximizing Coffee Profits

1. Jun 30, 2009

### jheld

1. The problem statement, all variables and given/known data
Mocha beans are priced at X dollars per pound, and Kona beans at y dollars per pound.
80 - 100x + 40y pounds of Mocha beans sold each week
20 + 60x -35y Kona beans sold each week. Cost of the beans is $2 per lbs of Mocha and$4 of Kona beans to the owners.
How should the owners price the coffee beans in order to maximize their profits?

2. Relevant equations
Looking for critical points (maximums, extremum).
Take the first derivative of each partial, see what the variable should be for it to be equal to 0.
Then go to the second partials and put them in the Hessian matrix, and solve with the derivative to see what the max should be.

3. The attempt at a solution
Solving for Mocha first..
f_x = -100
f_y = 40

5. Jul 5, 2009

### jheld

Okay, so far I understand. And, just to clarify, I wasn't using f_x and f_y notation for the hell of it, I was just making f the general equation. But, I understand the x - 2 and y -4 in conjunction with the two equations given.

So, I have taken the first (and second order) derivatives of each equation, but from there I am a little stuck. I have a few different points, and thus I am unsure of where they connect.

P_M(x) = -120 - 200x + 40y, where x = 0, y = 3;
P_M(y) = 40x - 80, where x = 2, y = 0;

P_K(x) = 60y - 240, where y = 4;
P_K(y) = 60x - 70y + 160, where x = -8/3, y = 16/7

I have also calculated the 2nd order partial derivatives, but I wanted to make sure that this was on the right track.

6. Jul 6, 2009

### compliant

It's on the right track, but to finish it off, you should also apply the second derivative test for partial deriviatives.