Consider the function of two variables f(x,y)=y 3 + x 2 - 6xy A) Calculate the f
ID: 2870794 • Letter: C
Question
Consider the function of two variables f(x,y)=y3 + x2- 6xy
A) Calculate the first order partial derivatives:
f/x=
f/y=
B) Find two candidates for maxima and minima by seting thepartials in part (a) both equal to zero and solving the resulting system for substitution.
C) Calculate the second order partial derivatives,
2f/x2=
2f/y2=
2f/xy=
D) Calculate D = Determinant of the Hessian matrix at each of the candidates you found and use it to classify each candidate as a Local Maximum, Local Minimum, or Saddle Point
Explanation / Answer
Consider the function of two variables f(x,y)=y3 + x2- 6xy
A) Calculate the first order partial derivatives:
f/x=2x-6y
f/y=3y^2 -6x
B) Find two candidates for maxima and minima by seting thepartials in part (a) both equal to zero and solving the resulting system for substitution.
2x-6y=0==>x-3y=0==>x=3y
,3y^2 -6x=0==>2x-y^2 =0==>2*3y -y^2 =0==>6y -y^2 =0==>y(6-y)=0 ==>y= 0,y=6
y=0==>x=0 ,y=6==>x=18
==>(0,0),(18,6) are critical points
C) Calculate the second order partial derivatives,
2f/x2=2
2f/y2=6y
2f/xy=0
D) Calculate D = Determinant of the Hessian matrix at each of the candidates you found and use it to classify each candidate as a Local Maximum, Local Minimum, or Saddle Point
for (0,0) 2f/x2=2
2f/y2=6*0=0
2f/xy=0
D=2*0-0 =0 inconclusive we need second derivative test
for (18,6)
for (0,0) 2f/x2=2
2f/y2=6*6=36
2f/xy=0
D=2*36-0=72>0
2f/x2 >0,D>0 ==>(18,6) is local minimum