2.10 Lagrange Multipliers

The surface $S$ is given by the equation $g(x,y,z)=0$. You are walking on $S$ measuring the function $f(x,y,z)$ as you go. You are currently at the point $(x_0,y_0,z_0)$ where $f$ takes its largest value on $S$, and are walking in the direction $\vd\ne\vZero$. Because you are walking on $S$, the vector $\vd$ is tangent to $S$ at $(x_0,y_0,z_0)$.

Find the maximum and minimum values of the function
$f(x,y,z)=x+y-z$ on the sphere $x^2+y^2+z^2=1$.

Find $a,\ b$ and $c$ so that the volume $\frac{4\pi}{3} abc$ of an ellipsoid $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ passing through the point $(1,2,1)$ is as small as possible.

Use the Method of Lagrange Multipliers to find the minimum value of $z = x^2 + y^2$ subject to $x^2 y = 1$. At which point or points does the minimum occur?

Use the Method of Lagrange Multipliers to find the radius of the base and the height of a right circular cylinder of maximum volume which can be fit inside the unit sphere $x^2 + y^2 + z^2 = 1$.

Use the method of Lagrange Multipliers to find the maximum and minimum values of

subject to the constraint

Find the maximum and minimum values of $f(x,y) = x^2 + y^2$ subject to the constraint $x^4 + y^4 = 1$.

Use Lagrange multipliers to find the points on the sphere $z^2 + x^2 + y^2 - 2y - 10 = 0$ closest to and farthest from the point $(1, -2, 1)$.

Use Lagrange multipliers to find the maximum and minimum values of the function $f(x,y,z) = x^2 + y^2 -\frac{1}{20} z^2$ on the curve of intersection of the plane $x + 2y + z = 10$ and the paraboloid $x^2 + y^2 - z = 0$.

Find the point $P = (x, y, z)$ (with $x$, $y$ and $z> 0$) on the surface $x^3 y^2 z = 6 \sqrt{3}$ that is closest to the origin.

Find the maximum value of $f (x, y, z) = xyz$ on the ellipsoid

Specify all points at which this maximum value occurs.

Find the radius of the largest sphere centred at the origin that can be inscribed inside (that is, enclosed inside) the ellipsoid

Let $C$ be the intersection of the plane $x + y + z = 2$ and the sphere $x^2 + y^2 + z^2 = 2$.

Use Lagrange multipliers to find the minimum and maximum values of $(x + z)e^y$ subject to $x^2 + y^2 + z^2 = 6$.

Find the points on the ellipse $2x^2 + 4xy + 5y^2 = 30$ which are closest to and farthest from the origin.

Find the ends of the major and minor axes of the ellipse $3x^2-2xy+3y^2=4$.

A closed rectangular box with a volume of 96 cubic meters is to be constructed of two materials. The material for the top costs twice as much per square meter as that for the sides and bottom. Use the method of Lagrange multipliers to find the dimensions of the least expensive box.

Consider the unit sphere

in $\bbbr^3$. Assume that the temperature at a point $(x,y,z)$ of $S$ is

Find the hottest and coldest temperatures on $S$.

Find the dimensions of the box of maximum volume which has its faces parallel to the coordinate planes and which is contained inside the region $0\le z\le 48-4x^2-3y^2$.

A rectangular bin is to be made of a wooden base and heavy cardboard with no top. If wood is three times more expensive than cardboard, find the dimensions of the cheapest bin which has a volume of $12{\rm m}^3$.

A closed rectangular box having a volume of $4$ cubic metres is to be built with material that costs $8 per square metre for the sides but $12 per square metre for the top and bottom. Find the least expensive dimensions for the box.

Suppose that $a$, $b$, $c$ are all greater than zero and let $D$ be the pyramid bounded by the plane $ax+by+cz=1$ and the 3 coordinate planes. Use the method of Lagrange multipliers to find the largest possible volume of $D$ if the plane $ax + by + cz = 1$ is required to pass through the point $(1, 2, 3)$. (The volume of a pyramid is equal to one-third of the area of its base times the height.)

Use Lagrange multipliers to find the minimum distance from the origin to all points on the intersection of the curves

Find the largest and smallest values of

on the sphere $x^2 + y^2 + z^2 = 36$. Determine all points at which these values occur.

The temperature in the plane is given by $T(x,y) = e^y\big(x^2+y^2\big)$.

Let $f(x,y)$ have continuous partial derivatives. Consider the problem of finding local minima and maxima of $f(x,y)$ on the curve $xy=1$.

Show that (E1) is equivalent to (E2), in the sense that