Find all points where the given function has any local extrema. Give the values

Question

Find all points where the given function has any local extrema. Give the values of any local extrema. Identify any saddle points. f(x,y) = x^2 + xy + y^2 + 6x - 3y + 3 Find the local maxima. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. There are local maxima located at (Simplify your answers. Type ordered pairs. Use a comma to separate answers as needed.) There are no local maxima. Find the values of the local maxima. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. The values of the local maxima are (Simplify your answers. Use a comma to separate answers as needed.) There are no local maxima. Find the local minima. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. There are local minima located at (Simplify your answers. Type ordered pairs. Use a comma to separate answers as needed.) There are no local minima. Find the values of the local minima. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. The values of the local minima are (Simplify your answers. Use a comma to separate answers as needed.) There are no local minima. Find the saddle points. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. There are saddle points located at (Simplify your answers. Type ordered pairs. Use a comma to separate answers as needed.) There are no saddle points.

Explanation / Answer

f(x,y)=x2+xy+y2+6x-3y+3

fx=0==>2x+y+0+6 =0==>2x+y=-6 ==>y=-6-2x

fy=0==>0+x+2y-3 =0==>x+2y=3

x+2y=3

==>x+2(-6-2x)=3

==>x-12-4x=3

==>-3x=15

==>x=-5

x+2y=3 ==>-5+2y=3 ==>y=4

(x,y)=(-5,4) is the only critical point

fxx=2, fyy=2,fxy=1

D=fxxfyy-(fxy)2

D=2*2-(1)2 =3>0

D>0,fxx>0==>(x,y)=(-5,4) is the local minimum point

f(-5,4)=(-5)2+(-5)(4)+42+6(-5)-3*4+3

f(-5,4)=25 -20+16-30-12+3

local minimum value =f(-5,4)=-18

no local maximum or saddlepoints

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