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

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

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

extreme points

f (x) = 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)

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