What is the extreme f(x,y)=(4x^2+10y^2)e^{-y} ?

The extreme f(x,y)=(4x^2+10y^2)e^{-y} is Minimum(0,0),Saddle(0,2)

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extreme f(x,y)=(4x^2+10y^2)e^{-y}

extreme points

f (x, y) = (4 x^{2} + 10 y^{2}) e^{- y}

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

Solution steps

Find the critical points:

(0, 0), (0, 2)

\frac{\partial ^{2} f}{\partial x ^{2}} = 2 (5 e^{- y} y^{2} - 20 e^{- y} y + 2 e^{- y} x^{2} + 10 e^{- y})

D (x, y) = - 32 e^{- 2 y} x^{2} + 160 e^{- 2 y} + 80 e^{- 2 y} y^{2} - 320 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)

P (x, y) = 6 x^{2} - x y - y^{2} + y + 8 x + 2

f (x) = (x + 2)^{\frac{4}{3}}

f (x) = x^{3} - 3 x^{2} + 1, - \frac{1}{2} \leq x \leq 4

f (x) = \frac{x ^{2} - 10 x + 25}{x - 10}

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

What is the extreme f(x,y)=(4x^2+10y^2)e^{-y} ?
The extreme f(x,y)=(4x^2+10y^2)e^{-y} is Minimum(0,0),Saddle(0,2)