Question

6. Consider the sphere S cut out by z2 + y2 22. Maximize (Daf)P where y, z) 2y +3z and u is a unit vector in the tangent plane to S at the point (A) v3 (E) 2v3 (B) 1+2V2 (C) 2 v3 (G) 3/2 (D) V2

Answer 1

f(x,y,z)--x + 2y 3z First we find the normal direction to the surface S. For this Consider F(x,y, z)-x2 + y2 + Z2-2-0 At the point (1, 0,1) Thus the tangent plane to the sphere at the point P-(1,0,1) is given by or, 2(x - 1) + 2(z -1)-0 or,x z 2... ) Any vector in the plane() is given by

u = (x,y, 2-х) And unit vector is given by (x,y, 2-х) f(x,y,z) -x2y 3z Vf = (-1,23) VfOP (-1,2 3) =(-1,23).一一(x,y, 2-х) x2 + y2 + (2 - x)2 -x 2y 6-3x -4x +2y +6 x2 + y2 + (2-x)2 At P(1,0 1) -4+06 12 +0 (2-1)2 V2 Thus the answer is /2

Answer D.

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