Question

Determinar os Pontos Críticos da função e classifique esses pontos:
F(X,Y)= 2x^2+2y^2-2xy^2+4x+4=0

Answer 1

$2x^2+2y^2-2xy^2+4x+4=0\\\\4x+4yy'-2(y^2+2yy'x)+4=0\\\\4x+4yy'-2y^2-4xyy'+4=0\\\\4yy'-4xyy'=-4x+2y^2-4\\\\y'=\dfrac{2y^2-4x-4}{4y-4xy}\\\\\\\dfrac{2y^2-4x-4}{4y-4xy}=0\\\\2y^2-4x-4=0\\\\2y^2=4x+4\\\\y^2=2x+2\\\\y=\pm\sqrt{2x+2}$$2x^2+2y^2-2xy^2+4x+4=0\\\\2x^2+2(2x+2)-2x(2x+2)+4x+4=0\\\\2x^2+4x+4-4x^2-4x+4x+4\\\\-2x^2+4x+8=0\\\\x^2-2x-4=0\\\\x=1\pm\sqrt{5}$