- Maximize x + y + z subject to the constraint x
^{2}+ y^{2}+ z^{2}= 3. - Minimize 2x − y + z subject to the constraint 2x
^{2}+ y^{2}+ z^{2}= 36. - Minimize x + 3y −z subject to the constraint 2x
^{2}+ y^{2}− 2z^{2}= 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 +≤ 1.
- Maximize x + 2y subject to the condition that x
^{2}+≤ 1. - Maximize x + y subject to the condition that x
^{2}++ z^{2}≤ 1. - Minimize x + y + z subject to the condition that x
^{2}++ z^{2}≤ 1. - Find the points on y
^{2}x = 16 which are closest to. - Find the points on y
^{2}x = 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=1}^{n}x_{j}subject to the constraint ∑_{j=1}^{n}x_{j}^{2}= a^{2}. Your answer should be some function of a which you may assume is a positive number. - Find the point on the level surface 4 x
^{2}+ y^{2}− z^{2}= 1which is closest to. - A curve is formed from the intersection of the plane, 2x+y+z = 3 and the cylinder x
^{2}+y^{2}= 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
x
^{2}+ y^{2}+ z^{2}= 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 A
_{ij}= A_{ji}and recall_{i}= A_{ij}x_{j}where as usual, sum over the repeated index. Show that= 2 A_{ij}x_{j}. Show that when you use the method of Lagrange multipliers to maximize the function A_{ij}x_{j}x_{i}subject to the constraint, ∑_{j=1}^{n}x_{j}^{2}= 1, the value of λ which corresponds to the maximum value of this functions is such that A_{ij}x_{j}= λx_{i}. Thus Ax = λx. Thus λ is an eigenvalue of the matrix A. - Here are two lines.
and x =

^{T}. Find points p_{1}on the first line and p_{2}on the second with the property thatis 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 it.- Find the point on the intersection of z = x
^{2}+ y^{2}and x + y + z = 1 which is closest to. - Minimize xyz subject to the constraints x
^{2}+ y^{2}+ z^{2}= r^{2}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 2x
^{2}+ xy + z^{2}= 16 which is closest to. - Find the point on x
^{2}+ y^{2}+ z^{2}= 1 closest to the plane x + y + z = 10. - Find the point on ++ z
^{2}= 1 closest to the plane x + y + z = 10. - Let x
_{1},,x_{5}be 5 positive numbers. Maximize their product subject to the constraint that - Let f= x
_{1}^{n}x_{2}^{n−1}x_{n}^{1}. Then f achieves a maximum on the set S ≡If x ∈ S is the point where this maximum is achieved, find x

_{1}∕x_{n}. ^{∗}Letbe a point on the ellipse, x^{2}∕a^{2}+ y^{2}∕b^{2}= 1 which is in the first quadrant. Extend the tangent line throughtill it intersects the x and y axes and let Adenote 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 ∏
_{i=1}^{n}x_{i}^{2}subject to the constraint, ∑

_{i=1}^{n}x_{i}^{2}= r^{2}. Show that the maximum is^{n}. Now show from this thatand finally, conclude that if each number x

_{i}≥ 0, thenand there exist values of the x

_{i}for which equality holds. This says the “geometric mean” is always smaller than the arithmetic mean. - Maximize x
^{2}y^{2}subject to the constraintwhere p,q are real numbers larger than 1 which have the property that

show that the maximum is achieved when x

^{2p}= y^{2q}and equals r^{2}. Now conclude that if x,y > 0, thenand there are values of x and y where this inequality is an equation.

- The area of the ellipse x
^{2}∕a^{2}+ y^{2}∕b^{2}≤ 1 is πab which is given to equal π. The length of the ellipse is ∫_{0}^{2π}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 x
^{2}+ x + y^{2}− 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 C^{1}function. The method does not apply to the corners of this region, the pointsand. Therefore, you need to consider these points also.

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