- Find the points where possible local minima or local maxima occur in the following functions.
- x2 − 2x + 5 + y2 − 4y
- −xy + y2 − y + x
- 3x2 − 4xy + 2y2 − 2y + 2x
- cos + sin
- x4 − 4x3y + 6x2y2 − 4xy3 + y4 + x2 − 2x
- y2x2 − 2xy2 + y2
- Find the volume of the largest box which can be inscribed in a sphere of radius a.
- Find in terms of a,b,c the volume of the largest box which can be inscribed in the ellipsoid
- Find three numbers which add to 36 whose product is as large as possible.
- Find three numbers x,y,z such that x2 + y2 + z2 = 1 and x + y + z is as large as possible.
- Find three numbers x,y,z such that x2 + y2 + z2 = 4 and xyz is as large as possible.
- 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
- 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
- The function f = 2
x2 + y2 is defined on the disk x2 + y2 ≤ 1. Find its maximum
- Find the point on the surface z = x2 + y + 1 which is closest to .
- Let L1 = and
L2 = 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.
Show that ∇f =
0. Now show that if is any nonzero unit vector, the function
t → f has a local minimum of 0 when
t = 0. Thus in every direction, this function
has a local minimum at but the function
f does not have a local minimum at