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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