Question as is
A grocery store carries two brands of frozen apple juice, a local brand that it obtains at the cost of 22 cents per can and a national brand that it obtains at the cost of 60 cents per can. The grocer estimates that if the local brad is sold for x cents per can and the national brand for y cents per can, approximately 70 - 5x+4y cans of the local brand and 80+6x-7y cans of the national brand will be sold each day. How should the grocer price each brand to maximize the profit from the sale of juice.
So they've given us the cost of each brand and the demand function for each. The price for one effecting the other. I figured the first thing I should do is find the Total Revenue function with respect to both x and y. So I got
R(x,y) = x(70-5x+4y) + y(80+6x-7y)
I distribute and combined like terms.
R(x,y) = -5x2+70x-7y2-80y+10xy
now I think I need to take the partial derivative of the function with respect to x then again with respect for y then plug in the prices of .22 and .60 for x and y.
Is this correct? Or do I need to take the second derivative of either of the partial derivatives then plug in the price?