What is the extreme f(x,y)=x^y ?

The extreme f(x,y)=x^y is Saddle(1,0)

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extreme f(x,y)=x^y

extreme points

f (x, y) = x^{y}

Saddle (1, 0)

Solution steps

Find the critical points:

(1, 0)

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

D (x, y) = y^{2} x^{2 y - 2} ln^{2} (x) - y x^{2 y - 2} ln^{2} (x) - x^{2 (y - 1)} - 2 y x^{2 y - 2} ln (x) - y^{2} x^{2 (y - 1)} ln^{2} (x)

Check the sign of at each critical point

Check critical point

(1, 0) :

Saddle

Saddle (1, 0)

f (x) = 4 x^{3} - 39 x^{2} + 90 x + 2

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

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

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

f (x) = cos (3 x)