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

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

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

extreme points

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

Minimum (1, 2), Saddle (- 1, 2), Saddle (1, - 2), Maximum (- 1, - 2)

Solution steps

Find the critical points:

(1, 2), (- 1, 2), (1, - 2), (- 1, - 2)

\frac{\partial ^{2} f}{\partial x ^{2}} = \frac{8}{y ^{3}}

D (x, y) = \frac{32}{x ^{3} y ^{3}}

Check the sign of at each critical point

Check critical point

(1, 2) :

Minimum

Check critical point

(- 1, 2) :

Saddle

Check critical point

(1, - 2) :

Saddle

Check critical point

(- 1, - 2) :

Maximum

Minimum (1, 2), Saddle (- 1, 2), Saddle (1, - 2), Maximum (- 1, - 2)

f (x, y) = - x - y + 2

f (x, y) = 18 x y - 3 x^{2} y - 2 x y^{2}

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

f (x, y) = x^{3} + y^{3} - 6 x y

f (x) = 2 x + cos (2 x)

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