What is the extreme f(x,y)= 1/x+xy+1/y ?

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

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extreme f(x,y)= 1/x+xy+1/y

extreme points

f (x, y) = \frac{1}{x} + x y + \frac{1}{y}

Minimum (1, 1)

Solution steps

Find the critical points:

(1, 1)

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

D (x, y) = \frac{4}{x ^{3} y ^{3}} - 1

Check the sign of at each critical point

Check critical point

(1, 1) :

Minimum

Minimum (1, 1)

f (x) = 2 x^{3} - 6 x^{2} - 18 x + 1

y = - 4 x^{3} + 60 x^{2} - 252 x + 8

f (x, y) = 10 - x^{2} y + 3 x y + x y^{2}

2 x^{3} - 9 x^{2} + 12 x

y = \frac{2 x}{x - 6}

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