Engineering Math

13.2 Exercises

1. Find the points where possible local minima or local maxima occur in the following functions.
   1. x² − 2x + 5 + y² − 4y
   2. −xy + y² − y + x
   3. 3x² − 4xy + 2y² − 2y + 2x
   4. cos
      (x)
      + sin
      (2y)
   5. x⁴ − 4x³y + 6x²y² − 4xy³ + y⁴ + x² − 2x
   6. y²x² − 2xy² + y²
2. Find the volume of the largest box which can be inscribed in a sphere of radius a.
3. Find in terms of a,b,c the volume of the largest box which can be inscribed in the ellipsoid
   x2 a2
   +
   y2- b2
   +
   z2 c2
   = 1.
4. Find three numbers which add to 36 whose product is as large as possible.
5. Find three numbers x,y,z such that x² + y² + z² = 1 and x + y + z is as large as possible.
6. Find three numbers x,y,z such that x² + y² + z² = 4 and xyz is as large as possible.
7. A feeding trough in the form of a trapezoid with equal base angles is made from a long rectangular piece of metal of width 24 inches by bending up equal strips along both sides. Find the base angles and the width of these strips which will maximize the volume of the feeding trough.
   PICT

8. An open box (no top) is to contain 40 cubic feet. The material for the bottom costs twice as much as the material for the sides. Find the dimensions of the box which is cheapest.
9. The function f
   (x,y)
   = 2x² + y² is defined on the disk x² + y² ≤ 1. Find its maximum value.
10. Find the point on the surface z = x² + y + 1 which is closest to
    (0,0,0)
    .
11. Let L₁ =
    (t,2t,3 − t)
    and L₂ =
    (2s,s+ 2,4− s)
    be two lines. Find a pair of points, one on the first line and the other on the second such that these two points are closer together than any other pair of points on the two lines.
12. ^∗Let

    ( |{ − 1 if y = x2,x ⁄= 0 f (x,y) = (y − x2)2 if y ⁄= x2 |( 0 if (x,y) = (0,0)

    Show that ∇f

    (0,0)
    = 0. Now show that if
    (a,b)
    is any nonzero unit vector, the function t → f
    (ta,tb)
    has a local minimum of 0 when t = 0. Thus in every direction, this function has a local minimum at
    (0,0)
    but the function f does not have a local minimum at
    (0,0)
    .