Find all locations and values of the local maxima and minima and the location of
ID: 2883883 • Letter: F
Question
Find all locations and values of the local maxima and minima and the location of the saddle points of the given function. f(x, y) = x^3 + y^3 + 3x^2 - 6y^2 - 3 Select the correct choice below and fill in an answer boxes within your choice. A. There are local maxima located at (Simplify your answer. Type an ordered pair Use a comma to separate answers as needed.) B. There are no local maxima. Select the correct choice below and fill in any answer boxes within your choice. A. The values of the local maxima are (Simplify your answer. Use a comma to separate answers as needed.) B. There are no local maxima. Select the correct choice below and fill in any answer boxes within your choice. A. There are local minima located at (Simplify your answer. Type an ordered pair Use a comma to separate answers as needed.) B. There are no local minima Select the correct choice below and fill in any answer boxes within your choice. A. The values of the local minima are (Simplify your answer. Use a comma to separate answers as needed.) B. There are no local minima. Select the correct choice below and fill in an answer boxes within your choice. A. There are saddle points located at (Simplify your answer. Type an ordered pair Use a comma to separate answers as needed.) B. There are no saddle points.Explanation / Answer
f(x,y)=x3+y3+3x2-6y2-3
fx=3x2+0+6x-0-0,fy=0+3y2+0-12y-0
fx=3x2+6x,fy=3y2-12y
for critical points fx=0,fy=0
3x2+6x=0,3y2-12y=0
3x(x+2)=0,3y(y-4)=0
x=0,x=-2,y=0,y=4
critical points are (0,0),(0,4),(-2,0),(-2,4)
fx=3x2+6x,fy=3y2-12y
fxx=6x+6,fyy=6y-12,fxy=0,D=fxxfyy-fxy2
at (0,0), fxx=0+6,fyy=0-12,fxy=0,D=-72<0
at (0,4),fxx=6 >0,fyy=12,fxy=0,D=72>0
at (-2,0),fxx=-6 <0,fyy=-12,fxy=0,D=72>0
at (-2,4),fxx=-6 <0,fyy=12,fxy=0,D=-72<0
f(0,4)=-35,f(-2,0)=1
there are local maxima located at (-2,0)
the values of local maxima are 1
there are local minima located at (0.4)
the values of local minima are -35
there are saddle points located at (0,0),(-2,4)
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