What is the extreme f(x)=x^2+y^2+2/(xy) ?

The extreme f(x)=x^2+y^2+2/(xy) is Minimum(1,1),Minimum(-1,-1)

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extreme f(x)=x^2+y^2+2/(xy)

extreme points

f (x) = x^{2} + y^{2} + \frac{2}{x y}

Minimum (1, 1), Minimum (- 1, - 1)

Solution steps

Find the critical points:

(1, 1), (- 1, - 1)

\frac{\partial ^{2} f}{\partial x ^{2}} = 2 + \frac{4}{x y ^{3}}

D (x, y) = 4 + \frac{8}{x y ^{3}} + \frac{8}{x ^{3} y} + \frac{12}{x ^{4} y ^{4}}

Check the sign of at each critical point

Check critical point

(1, 1) :

Minimum

Check critical point

(- 1, - 1) :

Minimum

Minimum (1, 1), Minimum (- 1, - 1)

f (x, y) = - 14 + 5 \cdot x^{2} + x \cdot y + y^{2}

f (x) = 5 x + \frac{3}{x}

f (x y) = x y e^{- x^{2} - y^{2}}

f (x, y) = x^{2} + y^{2} - 3 x y

f (x, y) = x^{4} + y^{4} - 4 x y + 4

What is the extreme f(x)=x^2+y^2+2/(xy) ?
The extreme f(x)=x^2+y^2+2/(xy) is Minimum(1,1),Minimum(-1,-1)