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

How do you optimize $f\left(x,y\right)=xy-{x}^{2}+{e}^{y}$ subject to $x-y=8$?
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dominicsheq8
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 $\frac{d}{dy}$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).
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