Question

How do you optimize `f(x,y)=xy-x^2+e^y` subject to `x-y=8`?

Answer 1

Minimum of f(x, y) = f(3 ln 2 + 8, 3 ln 2 ).
= `-`8 ( 3 ln 2 + 7 )
= `-`72.6355, nearly.

Explanation:

Substitute x = y + 8.
f(x, y) = g(y) = y(y + 8) `-`(y + 8)^2 + e^y = `-`8 ( y + 8 ) + e^y..
Necessary condition for g(y) to be either a maximum or a minimum is `d/dy`g(y) = 0. This gives y = 3 ln 2#.
The second derivative is e^y > 0, for all y. This is the sufficient condition that g(3 ln 2) is the minimum of g(y).