13.10.1.a. Find the point P(x, y, z) on the plane 2x + y Z 5 = 0 that is closest to the origin. 5 s 5 . . s ANS. P(,g,g), distances?g 13.10.1.b. Find the point P(x,y,z) on the plane x + y + z = 3 that is closest to the origin. ANS: P(1,1,1), distance is V3 13.10.l.c. Find the point P(x, y, z) on the plane ax + by + c2 = d that is closest to the origin. Answer is omitted. 13.102 Extrema on an ellipse. Find the points on the ellipse x2 + Zyz = 1 where f(x,y) = xy has its extreme values. (assesi) 13.103 Minimum surface area with fixed volume. Find the dimensions of the closed right circular cylindrical can of smallest surface area whose volume is 167: cm3. ANS: Answer Omitted 13.10.4 Rectangle of greatest area in an ellipse. Use the method of Lagrange multipliers to find the 2 2 dimensions of the rectangle of greatest area that can be inscribed in the ellipse 16 + y? = 1 with sides parallel to the coordinate axes. ANS: Length = 4J2, Width = so 13.10.35 Extrema on a circle. Find the maximum and minimum values of):2 + y2 subject to the constraint x2 2x + y: 43/ = 0. ANS: f(0,0) = 0 is minimum; f(2,4) = 20 is maximum 13.106 Extrema on a sphere. Find the points on the sphere x2 + y2 + 22 = 30 that maximize and minimize f(x, 3/, z). f(x,y,z) = x 23/ + 52 ANS: f(1,2, 5) = 30 is maximum; f(1, 2, 5) = 30 is minimum