What is the extreme f(x,y)=(5x^2+7y^2)e^{-y} ?

The extreme f(x,y)=(5x^2+7y^2)e^{-y} is Minimum(0,0),Saddle(0,2)

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extreme f(x,y)=(5x^2+7y^2)e^{-y}

extreme points

f (x, y) = (5 x^{2} + 7 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}} = - e^{- y} (- 7 y^{2} + 28 y - 5 x^{2} - 14)

D (x, y) = - 50 e^{- 2 y} x^{2} + 140 e^{- 2 y} + 70 e^{- 2 y} y^{2} - 280 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)

8 - 4 x

f (x) = x^{2} - x - 2, 0 \leq x \leq 2

- x^{3} + 12 x - 19

f (x) = 68 - x^{2} - y^{2}

y = \frac{x ^{2}}{x ^{2} + 243}

What is the extreme f(x,y)=(5x^2+7y^2)e^{-y} ?
The extreme f(x,y)=(5x^2+7y^2)e^{-y} is Minimum(0,0),Saddle(0,2)