First, take the equation:

x(x-15) + y(y-10) + 78 <= 0

and complete the square. You should end up with an equation:

(x-a)^2 + (y-b)^2 <= r^2

where a, b, and r are constants.

Now how much calculus do you know? Since you are finding min and max's, I have to assume you know some, but this seems like a Multivariable Calculus question. Is this what you're taking?

If so, all you need to do is derivative with respect to x and y, find the zeros and the end points on the region. Then find which one of these is the highest and which is the lowest.