What is the extreme f(x,y)=xln(y)+ye^x-x^2 ?

The extreme f(x,y)=xln(y)+ye^x-x^2 is Saddle(-(\W_{0}(4))/4 ,(e^{(\W_{0}(4))/4}\W_{0}(4))/(4))

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extreme f(x,y)=xln(y)+ye^x-x^2

extreme

Saddle (- \frac{W _{0} ( 4 )}{4}, \frac{e ^{\frac{W _{0} ( 4 )}{4}} W _{0} ( 4 )}{4})

Solution steps

Find the critical points:

(- \frac{W _{0} ( 4 )}{4}, \frac{e ^{\frac{W _{0} ( 4 )}{4}} W _{0} ( 4 )}{4})

\frac{\partial ^{2} f}{\partial x ^{2}} = - \frac{x}{y ^{2}}

D (x, y) = \frac{- e ^{x} x y + 2 x - 1 - 2 e ^{x} y - e ^{2 x} y ^{2}}{y ^{2}}

Check the sign of at each critical point

Check critical point

(- \frac{W _{0} ( 4 )}{4}, \frac{e ^{\frac{W _{0} ( 4 )}{4}} W _{0} ( 4 )}{4}) :

Saddle

Saddle (- \frac{W _{0} ( 4 )}{4}, \frac{e ^{\frac{W _{0} ( 4 )}{4}} W _{0} ( 4 )}{4})

What is the extreme f(x,y)=xln(y)+ye^x-x^2 ?
The extreme f(x,y)=xln(y)+ye^x-x^2 is Saddle(-(\W_{0}(4))/4 ,(e^{(\W_{0}(4))/4}\W_{0}(4))/(4))