18.6. EXERCISES 343

6. Maximize x+2y subject to the condition that x2 + y2

9 ≤ 1.

7. Maximize x+ y subject to the condition that x2 + y2

9 + z2 ≤ 1.

8. Minimize x+ y+ z subject to the condition that x2 + y2

9 + z2 ≤ 1.

9. Find the points on y2x = 16 which are closest to (0,0).

10. Find the points on√

2y2x = 1 which are closest to (0,0).

11. Find points on xy = 4 farthest from (0,0) if any exist. If none exist, tell why. Whatdoes this say about the method of Lagrange multipliers?

12. A can is supposed to have a volume of 36π cubic centimeters. Find the dimensionsof the can which minimizes the surface area.

13. A can is supposed to have a volume of 36π cubic centimeters. The top and bottom ofthe can are made of tin costing 4 cents per square centimeter and the sides of the canare made of aluminum costing 5 cents per square centimeter. Find the dimensions ofthe can which minimizes the cost.

14. Minimize and maximize ∑nj=1 x j subject to the constraint ∑

nj=1 x2

j = a2. Your answershould be some function of a which you may assume is a positive number.

15. Find the point (x,y,z) on the level surface 4x2+y2−z2 = 1which is closest to (0,0,0).

16. A curve is formed from the intersection of the plane, 2x+ y+ z = 3 and the cylinderx2 + y2 = 4. Find the point on this curve which is closest to (0,0,0).

17. A curve is formed from the intersection of the plane, 2x+3y+ z = 3 and the spherex2 + y2 + z2 = 16. Find the point on this curve which is closest to (0,0,0).

18. Find the point on the plane, 2x+3y+ z = 4 which is closest to the point (1,2,3).

19. Let A = (Ai j) be an n× n matrix which is symmetric. Thus Ai j = A ji and recall(Ax)i =Ai jx j where as usual, sum over the repeated index. Show that ∂

∂xk(Ai jx jxi) =

2Ai jx j. Show that when you use the method of Lagrange multipliers to maximizethe function Ai jx jxi subject to the constraint, ∑

nj=1 x2

j = 1, the value of λ whichcorresponds to the maximum value of this functions is such that Ai jx j = λxi. ThusAx= λx. Thus λ is an eigenvalue of the matrix A.

20. Here are two lines.x= (1+2t,2+ t,3+ t)T

and x= (2+ s,1+2s,1+3s)T . Find points p1 on the first line and p2 on the secondwith the property that |p1−p2| is at least as small as the distance between any otherpair of points, one chosen on one line and the other on the other line.

21. ∗ Find points on the circle of radius r for the largest triangle which can be inscribedin it.

22. Find the point on the intersection of z = x2 + y2 and x+ y+ z = 1 which is closest to(0,0,0).