Question

(a) Find and classify all of the critical points of the function X f(x, y, z) = (x2 +42 + x2)3/2 on the unit sphere. (b) Find and classify all of the critical points of the function f(x, y, z) = x sin(x2 + y2 +22) on the sphere of radius

Answer 1

#@ flig, Z) = 2 (2²+42 +223/2 on the cuitsphere - fx = (+y+z3j*%2_ Boe? (x2+y2+245/2 fy = -3ry ( 26² +9²7 225812 fz = -342 (++y+Z2j $12 on the sphere of unit rading, fre = 1 - 37%, fy = -3xy , fz 2 -347. . Critical points would satisfy these three equations Simultaneously 1. for=1-30²20) a= 2 153 fy = f2 =0 Thes, o to, 0,01, 6100,0) are the coffical pointe, 572 forme = -30(92+ y² + Z27-5/2_66 (x²+4²+ 225 +1531**+ y2+237772 fyny = -3y(+*+ 9 +2 2742 + 1582y (x²+ y2 + 25th f22 = ~34674747°+ z 2792 + 15 *y*%$}+y*+z" D(155,0,0) = frame (yo, 0,0) xfyy , 0, 0)- Shay (tas, 0,08 ] = 54.27>0 famato, 0, 0)=-01 +15*9 > 0 Thus it is a relative minima D11a,0,0) = favel-331060) fgy (+3,0,0).fboy (fe,0,0))* A = -54.27 co is a saddle Thus, it

# 6 fox, y, z) = sin(22+ y 24 22), on the sphere of for = sin(222fy 24 24+2x²6x (21247 ²7 27 on the off sphere of radius S/2, we have of x=t We can easily get frog and fz to be zero, This there are no critical points a please give uprote,