What is the extreme f(x,y)=ln(xy)+2x^2-xy-6x ?

The extreme f(x,y)=ln(xy)+2x^2-xy-6x is Saddle(3/2 , 2/3)

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extreme f(x,y)=ln(xy)+2x^2-xy-6x

extreme points

f (x, y) = ln (x y) + 2 x^{2} - x y - 6 x

Saddle (\frac{3}{2}, \frac{2}{3})

Solution steps

Find the critical points:

(\frac{3}{2}, \frac{2}{3})

\frac{\partial ^{2} f}{\partial x ^{2}} = - \frac{1}{y ^{2}}

D (x, y) = - \frac{- 1 + 4 x ^{2}}{x ^{2} y ^{2}} - 1

Check the sign of at each critical point

Check critical point

(\frac{3}{2}, \frac{2}{3}) :

Saddle

Saddle (\frac{3}{2}, \frac{2}{3})

y = (x - 4)^{2} - 8

f (x) = x + e^{- 3 x}

y = (6 x - 5) e^{6 - 2 x}

f (x, y) = 1 - x^{2} + 1 - y^{2}

- x e^{- \frac{x}{4}}

What is the extreme f(x,y)=ln(xy)+2x^2-xy-6x ?
The extreme f(x,y)=ln(xy)+2x^2-xy-6x is Saddle(3/2 , 2/3)