What is the extreme xye^{x+2y} ?

The extreme xye^{x+2y} is Maximum(-1,-1/2)

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extreme xye^{x+2y}

extreme points

x y e^{x + 2 y}

Maximum (- 1, - \frac{1}{2})

Solution steps

Find the critical points:

(- 1, - \frac{1}{2})

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

D (x, y) = x y (e^{x + 2 y} x + 2 e^{x + 2 y}) (4 e^{x + 2 y} y + 4 e^{x + 2 y}) - e^{2 (2 y + x)} (1 + x)^{2} (1 + 2 y)^{2}

Check the sign of at each critical point

Check critical point

(- 1, - \frac{1}{2}) :

Maximum

Maximum (- 1, - \frac{1}{2})

1 - 2 x^{2}

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

x^{4} - 8 x^{2} + 5

\frac{( x ^{2} - 9 )}{x ^{2} + 9}

f (x) = x^{\frac{1}{5}} (x^{2} - 9)