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

The extreme f(x,y)=e^{2x}(x+y^2+2y) is Minimum(1/2 ,-1)

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

extreme points

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

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

Solution steps

Find the critical points:

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

\frac{\partial ^{2} f}{\partial x ^{2}} = 2 e^{2 x}

D (x, y) = - 8 e^{4 x} y^{2} - 16 e^{4 x} y - 8 e^{4 x} + 8 e^{4 x} x

Check the sign of at each critical point

Check critical point

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

Minimum

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

f (x) = \frac{2 x ^{\frac{5}{2}}}{5} - \frac{2 x ^{\frac{3}{2}}}{3} - 1

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

5 x^{3} - 5 x^{2} - 5 x + 9

\frac{x ^{3}}{x ^{2} - 49}

x^{2} (1 - x)^{3}

What is the extreme f(x,y)=e^{2x}(x+y^2+2y) ?
The extreme f(x,y)=e^{2x}(x+y^2+2y) is Minimum(1/2 ,-1)