What is the extreme f(x,y)=x+4y+2/(xy) ?

The extreme f(x,y)=x+4y+2/(xy) is Minimum(2, 1/2)

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extreme f(x,y)=x+4y+2/(xy)

extreme points

f (x, y) = x + 4 y + \frac{2}{x y}

Minimum (2, \frac{1}{2})

Solution steps

Find the critical points:

(2, \frac{1}{2})

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

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

Check the sign of at each critical point

Check critical point

(2, \frac{1}{2}) :

Minimum

Minimum (2, \frac{1}{2})

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

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

f (x, y) = 2 y^{3} - 3 x y + x^{2} - 1

- \frac{cos ( 2 x )}{2} - 2 sin (x)

f (x) = 2 x^{3} - x^{2} - 4 x - 1

What is the extreme f(x,y)=x+4y+2/(xy) ?
The extreme f(x,y)=x+4y+2/(xy) is Minimum(2, 1/2)