LON-CAPA maxmin-3 x CCalculus question Chega x PF Implicit differentiaion usin x

Question

LON-CAPA maxmin-3 x CCalculus question Chega x PF Implicit differentiaion usin x G exponential Google Sear xt e LON-CAPA LON-CAPA He x Ian C fi X https://homework.math.pitt.edu 1%2fdefault 12 Q O E pitt.edu/res/pitt/athanas/c3-maxmin-2.problem?symb:-uploaded%2fpitt%2f682410109ecle055d6pittmath Timer Notes Evaluate Feedback Print Info Course Contents PROBLEM SET 05 maxmin-3 Determine the critical points of the function ,y) Submit Answer Tries 0/8 For each point you listed above, calculate D fafy fa, and in the last box of each line below put a: 1 for "local minimum". 2 for "local maximum" 3 for "saddle point" For the point Gr1,y1) Therefore, at this point (ri, ym), there is a (put in 1, 2, or 3) D- For the point y,) Therefore, at this point (m2, y,), there is a (put in 1, 2, or 3) For the point y') D- Therefore, at this point Gra, ya), there is a (put in 1, 2, or 3) Submit Answer Tries 0/8 Post Discussion Send Feedback jsMath 6:46 PM 10/16/2015

Explanation / Answer

f(x,y)=y^3 +3xy^2 +22x^2 -16x

fx=0==>3y^2 +44x -16 =0

fy=0 ==>3y^2 +6xy=0 ==>3y(y+2x)=0 ==> y=0 ,y =-2x

3y^2 +44x -16 =0 , y=0 ==>3*0^2 +44x -16 =0==>44x =16 ==>x=4/11

3y^2 +44x -16 =0 , y=-2x ==>3(-2x)^2 +44x -16 =0 ==>12x^2 +44x -16=0==>3x^2 +11x -4=0==>3x^2 +12x-x -4=0==>3x(x+4)-1(x+4)=0==>(3x-1)(x+4)=0==>x=-4,x=1/3

y=-2x ,x=-4 ==>y=8 ,

y=-2x ,x=1/3==>y=-2/3 ,

points are (x,y)=(4/11,0) ,

points are (x,y)=(4/11,0) ,(-4,8),(1/3,-2/3)

fxx=44 ,fyy=6y+6x ,fxy=6y

D=fxx*fyy-(fxy)2

(x,y)=(4/11,0) ==>D=44*24/11 -0^2 ==>D>0, fxx>0----------->local minimum 1

(x,y)=(-4,8) ==>D=44*24 -48^2 ==>D<0, fxx>0 ------------------------>saddle point 3

(x,y)=(1/3,-2/3) ==>D=44*(-2) -(-4)2 ==>D<0, fxx>0----------------->saddle point 3

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