Answer to Question #305438 in Math for Fresh

Question #305438

Classify the critical points of f(x,y)= 4+x^3 + y^3 -3xy


1
Expert's answer
2022-03-07T15:38:01-0500
"f(x,y)= 4+x^3 + y^3 -3xy"

"f_x=3x^2-3y"

"f_y=3y^2-3x"

Find the critical point(s)


"f_x=0""f_y=0"

Then


"3x^2-3y=0""3y^2-3x=0"

"y=x^2"

"x^4-x=0"

Critical points are "(0, 0), (1, 1)"


"f_{xx}=6x"

"f_{xy}=f_{yx}=-3"

"f_{yy}=6y"


"D=\\begin{vmatrix}\n f_{xx} & f_{xy} \\\\\n f_{yx} & f_{yy}\n\\end{vmatrix}=\\begin{vmatrix}\n 6x & -3 \\\\\n -3 & 6y\n\\end{vmatrix}=36xy-9"

Point "(0,0)"


"D(0,0)=36(0)(0)-9=-9<0"

Then "f(0,0)" is not a local maximum or minimum. Point "(0,0)" is a saddle point of "f."


Point "(1,1)"

"D(1,1)=36(1)(1)-9=27>0"

"f_{xx}(1,1)=6(1)=6>0"

Then "f(1,1)=3" is a local minimum.



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