13.9     Lagrange Multiplier

1. In each of the following problems, find the extreme(maximum and minimum) values of   subjected to the given constrained:

(i) subject to .

(ii) subject to .

(iii) subject to .

(iv) subject to .

(v) subjected to

(vi) subjected to .

(vii) subjected to .

Can you generalize this?

2. Find the points on the sphere that are closest and farthest from the point .

3. Determine the dimensions of  a rectangular box, open at the top having  volume pf 32 cubic feets and requiring the least amount

of material for its construction.

4. Suppose the cost of manufacturing a particular type of box is such that the base of the box costs three times as much per square

foot as the sides and top. Find the dimensions of the box that minimize the cost for a given volume.

5. Find the maximum and minimum of on the ellipse given by the intersection of the cylinder

and the plane .

6. The cone is cut by the plane in some curve . Find the point on that is closest to the origin.

7. In the following exercises find the extreme values of subjected to the two constraints.

(a) ; ; .

(b) ; ; .

(c) ; ;

(d) ; ; .

8. (a )Use Lagrange multipliers to find the highest and lowest points on the ellipse .

(b) Use Lagrange multipliers to show  that the rectangle with maximum area that has a given perimeter is a square.

(c) Use Lagrange multipliers to show that the triangle with maximum area that has a given perimeter is equilateral.

(d) Find the maximum and minimum volumes of a rectangular box whose surface area is \$1200 cm^2\$ and whose total

edge length is \$.

(e) Use Sage to ind the maximum of , subject to the constraint

.

9. Find the absolute maximum value and minimum value of the function on the ellipse .

Using ,

Substituting in , then  =>

Substituting in =>  => =>

The absolute maximum value is

The absolute minimum value is

10. Find the absolute maximum value of the function on the ellipse .

Using ,

Substituting in  => >

Substituting in => =>

Hence and

Now the absolute maximum value of the function is

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