Find all critical points of the following function and class

Find all critical points of the following function, and classify them with the second partials test. f (x, y) = x^2/2 + 2xy - 7x+ 3y^2/2 - 11y + 8 Find all critical point of the following function, and classify them with the second partials test. f (x, y) = - 3x^2 + 4xy - 32x - 3y^2/2 + 23y + 9

Solution

1)

f(x,y)=(1_/2)x² +2xy -7x +(3_/2)y² -11y +8

f_x =x +2y -7 +0-0+0,f_y=0+2x -0 +3y -11 +0

f_x =x +2y -7 ,f_y=2x +3y -11

for critical points f_x =0 ,f_y=0

x +2y -7=0 ,2x +3y -11=0

=>x +2y =7 ,2x +3y =11

=>2(x+2y) -(2x+3y) =(2*7) -11

=>2x+4y -2x-3y =14-11

=>y=3

x +2y =7 ,y=3

=>x +(2*3) =7

=>x+6=7

=>x=1

(x,y)=(1,3) is the critical point

f_x =x +2y -7 ,f_y=2x +3y -11

f_xx =1 ,f_yy=3,f_xy =2 ,D=f_xxf_yy-f_xy²

D=(1*3)-2² =-1 <0

f(1,3)=-12

so saddle point at (1,3,-12)

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1)

f(x,y)=-3x² +4xy -32x -(3_/2)y² +23y +9

f_x =-6x +4y -32 -0+0+0,f_y=0+4x -0 -3y +23 +0

f_x =-6x +4y -32 ,f_y=4x-3y+23

for critical points f_x =0 ,f_y=0

-6x +4y -32=0 ,4x-3y+23 =0

-3x +2y =16 ,4x-3y=-23

on solving x=-2, y=5

(x,y)=(-2,5) is the critical point

f_x =-6x +4y -32 ,f_y=4x-3y+23

f_xx =-6 ,f_yy=-3,f_xy =4 ,D=f_xxf_yy-f_xy²

D=(-6*-3)-4² = 2>0

D>0,f_xx<0

f(-2,5)=197_/2

so local maximum at (-2,5,197_/2)