Glenn Hopkins

2022-07-16

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

dominicsheq8

Beginner2022-07-17Added 15 answers

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).

= −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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which is definitely not $0$ as $(x,y)\to (0,0)$. Same can be stated for ${f}_{y}$. But how to proceed with the first part?