- 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
- Find the point on the level surface 4
x2 + y2 − z2 = 1which is closest
- 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
i = Aijxj where as usual, sum over the repeated index.
Aijxj. Show that when you use the method
of Lagrange multipliers to maximize the function Aijxjxi subject to the
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 A.
- 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
- ∗ Find points on the circle of radius r for the largest triangle which can be inscribed
- Find the point on the intersection of z = x2 + y2 and x + y + z = 1 which is closest
- 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 possible.
- 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
- Let f =
xn1. Then f achieves a maximum on the set
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
a and b.
- Maximize ∏
subject to the constraint, ∑
i=1nxi2 = r2. Show that the maximum is
Now show from this that
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, then
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 possible.
- 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 points