This document contains 16 multi-part math problems involving optimization of functions, geometry, and physics. The problems cover topics like finding extrema of functions, finding points on lines, maximizing areas of geometric shapes given constraints, minimizing materials needed to construct cylinders and fences, and finding positions of maximum or minimum values of physical quantities like force and illumination.

1. EXERCISE SET 3.6
For 1 and 2, find the extrema of the continuous function over the given intervals.
4 x 2 for 2 x
1. y (a) [-1,3] (b) [-1,7] (c) [-3,3]
x 2 for 2 x
x for 1 x
2. y 2
(a) [½,2½] (b) [-½,1½] (c) [½,4]
( x 2) for 1 x
For 3 and 4 find the coordinate of the point on the line that is nearest the given point.
3. line: y = x + 1, point: (3,2) 4. Line: y = -2x, point: (-1,-3)
5. A farmer wishes to fence off a rectangular pasture along a straight river, the side along the
river requiring no fence. He wants to divide the pasture in two using a fence through the
middle and perpendicular to the river. What is the maximum area of the pasture if he has 420
yards of fence?
6. Repeat #5 if the farmer decides to fence along the river and the dividing fence runs parallel
to the river.
7. It is desired to fence a rectangle, and to divide it into two parts by a fence parallel to one of
its sides. If the total area is 216 square meters, find the dimensions that will use the least
amount of fence.
8. It is desired to fence a rectangle, and to divide it into three parts by two fences parallel to
one of its sides. If the total area is 200 square yards, find the dimensions that will use the least
amount of fence.

2. 9. What is the least amount of sheet metal needed to make a closed cylindrical tank that will
hold 250π cubic meters?
10. What is the least amount of sheet metal needed to make a closed cylindrical tank that will
hold 1728 cubic feet? Assume the circular ends are cut from a square sheet and the corner
pieces are thrown away.
11. The vertex of an isosceles triangle is at (0,4). The other two vertices are on the curve y = x2
and are the same distance below (0,4). Find the largest area of such a triangle.
12. The vertex of an isosceles triangle is at (0,0). The other two vertices are on the curve
y = 16 - x2 and are the same distance above (0,0). Find the largest area of such a triangle.
13. A duck is on the bank of a 100 foot wide river. A tree is directly across the river on the
other bank. A piece of bread is 200 feet from the tree on the same riverbank. If the duck can
swim 4 ft/sec and walk 5 ft/sec, what is the path the duck should take to get to the bread as
soon as possible?
14. A duck is on the bank of a 100 foot wide river. A tree is directly across the river on the
other bank. A piece of bread is 120 feet from the tree on the same riverbank. If the duck can
swim 4 ft/sec and walk 5 ft/sec, what is the path the duck should take to get to the bread as
soon as possible?
15. The force between an electric current flowing in a circle of radius r and a magnet located x
Kx
meters away is given by F 5
. Find x for maximum force if r = 4 meters.
2 2 2
(x r )
16. The illumination at a point on the x-axis due to two light sources is given by
16 2
I 2
. Find x so that the illumination is least.
x (6 x ) 2