- Maximize x + y + z subject to the constraint x2 + y2 + z2 = 3.
- Minimize 2x − y + z subject to the constraint 2x2 + y2 + z2 = 36.
- Minimize x + 3y −z subject to the constraint 2x2 + y2 − 2z2 = 36 if possible. Note there is no
guaranty this function has either a maximum or a minimum. Determine whether there exists
a minimum also.
- Find the dimensions of the largest rectangle which can be inscribed in a circle of radius r.
- Maximize 2x + y subject to the condition that +
- Maximize x + 2y subject to the condition that x2 +
- Maximize x + y subject to the condition that x2 + +
z2 ≤ 1.
- Minimize x + y + z subject to the condition that x2 + +
z2 ≤ 1.
- Find the points on y2x = 16 which are closest to .
- Find the points on
y2x = 1 which are closest to .
- Find points on xy = 4 farthest from if any exist. If none exist, tell why. What does this
say about the method of Lagrange multipliers?
- A can is supposed to have a volume of 36π cubic centimeters. Find the dimensions of the can
which minimizes the surface area.
- A can is supposed to have a volume of 36π cubic centimeters. The top and bottom of the
can are made of tin costing 4 cents per square centimeter and the sides of the can are made
of aluminum costing 5 cents per square centimeter. Find the dimensions of the can which
minimizes the cost.
- Minimize and maximize ∑
j=1nxj subject to the constraint ∑
j=1nxj2 = a2. Your answer
should be some function of a which you may assume is a positive number.
- Find the point on the level surface 4
x2 + y2 − z2 = 1which is closest to .
- A curve is formed from the intersection of the plane, 2x+y+z = 3 and the cylinder x2+y2 = 4.
Find the point on this curve which is closest to .
- A curve is formed from the intersection of the plane, 2x + 3y + z = 3 and the sphere
x2 + y2 + z2 = 16. Find the point on this curve which is closest to .
- Find the point on the plane, 2x + 3y + z = 4 which is closest to the point .
- Let A = be an
n×n matrix which is symmetric. Thus Aij = Aji and recall
i = Aijxj
where as usual, sum over the repeated index. Show that
Aijxj. Show that
when you use the method of Lagrange multipliers to maximize the function Aijxjxi subject
to the constraint, ∑
j=1nxj2 = 1, the value of λ which corresponds to the maximum value of
this functions is such that Aijxj = λxi. Thus Ax = λx. Thus λ is an eigenvalue of the matrix
- Here are two lines.
and x =
T. Find points p1 on the first line and p2 on the second with the
property that is at least as small as the distance between any other pair of points, one
chosen on one line and the other on the other line.
- ∗ Find points on the circle of radius r for the largest triangle which can be inscribed in
- Find the point on the intersection of z = x2 + y2 and x + y + z = 1 which is closest to
- Minimize xyz subject to the constraints x2 + y2 + z2 = r2 and x − y = 0.
- Let n be a positive integer. Find n numbers whose sum is 8n and the sum of the squares is as small as
- Find the point on the level surface 2x2 + xy + z2 = 16 which is closest to .
- Find the point on x2 + y2 + z2 = 1 closest to the plane x + y + z = 10.
- Find the point on +
z2 = 1 closest to the plane x + y + z = 10.
- Let x1,
,x5 be 5 positive numbers. Maximize their product subject to the constraint
- Let f =
xn1. Then f achieves a maximum on the set S ≡
If x ∈ S is the point where this maximum is achieved, find x1∕xn.
- ∗ Let be a point on the ellipse,
x2∕a2 + y2∕b2 = 1 which is in the first quadrant. Extend the
tangent line through till it intersects the
x and y axes and let A denote the area of the
triangle formed by this line and the two coordinate axes. Find the minimum value of the area of this
triangle as a function of
a and b.
- Maximize ∏
subject to the constraint, ∑
i=1nxi2 = r2. Show that the maximum is
n. Now show from this
and finally, conclude that if each number xi ≥ 0, then
and there exist values of the xi for which equality holds. This says the “geometric mean” is always
smaller than the arithmetic mean.
- Maximize x2y2 subject to the constraint
where p,q are real numbers larger than 1 which have the property that
show that the maximum is achieved when x2p = y2q and equals r2. Now conclude that if x,y > 0,
and there are values of x and y where this inequality is an equation.
- The area of the ellipse x2∕a2 + y2∕b2 ≤ 1 is πab which is given to equal π. The length of the ellipse is
dt. Find a,b such that the ellipse having this volume is as short as
- Consider the closed region in the xy plane which lies between the curve y = and
y = 0. Find
the maximum and minimum values of the function x2 + x + y2 − y on this region. Hint: First
observe that there is a solution because the region is compact. Next look for candidates for the
extreme point on the interior. When this is done, look for candidates on the boundary. Note that the
boundary of the region does not come as the level surface of a C1 function. The method does not
apply to the corners of this region, the points and
. Therefore, you need to consider these