Page 213 - Schaum's Outline of Theory and Problems of Advanced Calculus

204 APPLICATIONS OF PARTIAL DERIVATIVES [CHAP. 8

MAXIMA AND MINIMA, LAGRANGE MULTIPLIERS
2 3
8.58. Find the maxima and minima of Fðx; y; zÞ¼ xy z subject to the conditions x þ y þ z ¼ 6, x > 0; y > 0,
z > 0. Ans. maximum value ¼ 108 at x ¼ 1; y ¼ 2; z ¼ 3

8.59. What is the volume of the largest rectangular parallelepiped which can be inscribed in the ellipsoid
2
2
ffiffiffi
p
2
x =9 þ y =16 þ z =36 ¼ 1? Ans.64 3
2
2
2
2
8.60. (a)Find the maximum and minimum values of x þ y subject to the condition 3x þ 4xy þ 6y ¼ 140.
(b)Give a geometrical interpretation of the results in (a).
Ans. maximum value ¼ 70, minimum value ¼ 20
8.61. Solve Problem 8.23 using Lagrange multipliers.
2 2 2
8.62. Prove that in any triangle ABC there is a point P such that PA þ PB þ PC is a minimum and that P is the
intersection of the medians.
2
2
8.63. (a)Prove that the maximum and minimum values of f ðx; yÞ¼ x þ xy þ y in the unit square 0 @ x @ 1,
0 @ y @ 1are 3 and 0, respectively. (b) Can the result of (a)be obtained by setting the partial derivatives
of f ðx; yÞ with respect to x and y equal to zero. Explain.
2
2
2
8.64. Find the extreme values of z on the surface 2x þ 3y þ z 12xy þ 4xz ¼ 35.
Ans. maximum ¼ 5, minimum ¼ 5
8.65. Establish the method of Lagrange multipliers in the case where we wish to find the extreme values of
Fðx; y; zÞ subject to the two constraint conditions Gðx; y; zÞ¼ 0, Hðx; y; zÞ¼ 0.

8.66. Prove that the shortest distance from the origin to the curve of intersection of the surfaces xyz ¼ a and
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
y ¼ bx where a > 0; b > 0, is 3 aðb þ 1Þ=2b.
2
2
2
ffiffiffi
p
8.67. Find the volume of the ellipsoid 11x þ 9y þ 15z 4xy þ 10yz 20xz ¼ 80. Ans. 64 2=3
APPLICATIONS TO ERRORS
8.68. The diameter of a right circular cylinder is measured as 6:0 0:03 inches, while its height is measured as
4:0 0:02 inches. What is the largest possible (a) error and (b)percent error made in computing the
3
volume? Ans.(a)1.70in , (b)1.5%
8.69. The sides of a triangle are measured to be 12.0 and 15.0 feet, and the included angle 60.08.If the lengths can
be measured to within 1% accuracy, while the angle can be measured to within 2% accuracy, find the
maximum error and percent error in determining the (a)area and (b) opposite side of the triangle.
2
Ans.(a)2.501 ft ,3.21%; (b)0.287 ft, 2.08%

MISCELLANEOUS PROBLEMS
8.70. If and are cylindrical coordinates, a and b are any positive constants, and n is a positive integer, prove
n
n
that the surfaces sin n ¼ a and cos n ¼ b are mutually perpendicular along their curves of intersec-
tion.
2
8.71. Find an equation for the (a)tangent plane and (b) normal line to the surface 8r ¼ at the point where
r ¼ 1, ¼ =4; ¼ =2; ðr; ; Þ being spherical coordinates.
p ffiffiffi p ffiffiffi
x 2=2 2=2
ffiffiffi
p
2
2
Ans: 4x ð þ 4 Þ y þð4 Þz ¼ 2 2; y z
2
4 ¼ þ 4 ¼ 4
2
ðaÞ
ðbÞ