What is the extreme f(x,y)=e^y(x^2+y^2) ?

The extreme f(x,y)=e^y(x^2+y^2) is Minimum(0,0),Saddle(0,-2)

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extreme f(x,y)=e^y(x^2+y^2)

extreme points

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

Minimum (0, 0), Saddle (0, - 2)

Solution steps

Find the critical points:

(0, 0), (0, - 2)

\frac{\partial ^{2} f}{\partial x ^{2}} = e^{y} (y^{2} + 4 y + x^{2} + 2)

D (x, y) = - 2 e^{2 y} x^{2} + 4 e^{2 y} + 2 e^{2 y} y^{2} + 8 e^{2 y} y

Check the sign of at each critical point

Check critical point

(0, 0) :

Minimum

Check critical point

(0, - 2) :

Saddle

Minimum (0, 0), Saddle (0, - 2)

f (x) = - 2 cos (2 x)

f (x) = 3 x, - 8 \leq x \leq 8

f (x) = x^{\frac{2}{3}} - 7

f (x) = x^{4} + 2 x^{3} - 2 x + 5

f (x, y) = 2 x^{2} + 4 y^{2} + \frac{1}{2}

What is the extreme f(x,y)=e^y(x^2+y^2) ?
The extreme f(x,y)=e^y(x^2+y^2) is Minimum(0,0),Saddle(0,-2)