Question #103439
Obtain and classify the relative extrema values of the function
F(x,y)=x³+y³-63(x+y)+12xy
1
Expert's answer
2020-02-20T09:39:05-0500

F(x,y)=x3+y363(x+y)+12xyF(x,y)=x^3+y^3-63(x+y)+12xy


0 It's a polynomial function with continuous partial derivatives of any order.


1 What are the critical points?


Fx=3x263+12yF^{'}_x=3x^2-63+12y

Fy=3y263+12xF^{'}_y=3y^2-63+12x

x221+4y=0x^2-21+4y=0

y221+4x=0y^2-21+4x=0


2 Can we solve the system?


4y=21x24y=21-x^2(21x2)22116+416x=0(21-x^2)^2-21*16+4*16*x=0

44142x2+x4336+64x=0441-42x^2+x^4-336+64x=0

10542x2+x4+64x=0105-42x^2+x^4+64x=0

x442x2+64x+105=0P(x)x^4-42x^2+64x+105=0 \coloneqq P(x)


105=1357105=1*3*5*7

P(1)0, P(1)=0, P(3)=0, P(3)0P(1)\neq0,~P(-1)=0,~P(3)=0,~P(-3)\neq0

P(5)=0, P(5)0, P(7)0, P(7)=0P(5)=0,~P(-5)\neq0,~P(7)\neq0,~P(-7)=0


Good, all roots (pairs of) are real (and symmetric).


x=7  x=1  x=3   x=5x=-7~~x=-1~~x=3~~~x=5

y=7  y=5     y=3   y=1y=-7~~y=5~~~~~y=3~~~y=-1


3 What does the second derivative test say?

D=FxxFxxFxyFxyD=F^{''}_{xx}*F^{''}_{xx}-F^{''}_{xy}*F^{''}_{xy}

Fxx=6xF^{''}_{xx}=6x

Fyy=6yF^{''}_{yy}=6y

Fxy=12F^{''}_{xy}=12

D=36xy144D=36xy-144


D(7,7)>0,  Fxx<0    local maximumD(-7, -7)>0,~~F^{''}_{xx} < 0 \implies \textnormal{local maximum}

D(1,5)<0                        saddle pointD(-1, 5)<0~~~~~~~~~~~~~~~~~~~~\implies \textnormal{saddle point}

D(3,3)>0,        Fxx>0    local minimumD(3, 3)>0,~~~~~~~~F^{''}_{xx}>0 \implies \textnormal{local minimum}

D(5,1)<0                        saddle pointD(5, -1)<0~~~~~~~~~~~~~~~~~~~~\implies \textnormal{saddle point}



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