Section 4.7
Example Problems
- Find a positive number such that the sum of the number and its reciprocal is as small as possible. (#4 pg 283)
- A
rancher who wishes to fence off a rectangular area finds that the fencing
in the east –west direction will require extra reinforcement owing to
strong prevailing winds. Because of
this, the cost of fencing in the east-west direction will be $12 per
(linear) yard, as opposed to a cost of $8 per yard for fencing in the
north-south direction. Find the
dimensions of the largest possible rectangular area that can be fenced for
$4800.
- Find
the dimensions of the rectangle of largest area that can be inscribed in
an equilateral triangle of side L if one side of the rectangle lies on the
base of the triangle. (#21 pg 284)
- A box
with a square base and open top must have a volume of
. Find the
dimensions of the box that minimize the amount of material used. (#10 pg
284)
- A piece of wire 10 m long is cut into two pieces. One piece is bent into a square and the other into a circle. How should the wire be cut so that the total area enclosed is (a) maximized (b) minimized? (#32 pg 284)
- A right circular cylinder is inscribed in a cone with height h and radius r. Find the largest possible volume of such a cylinder. (#26 pg 284)
- The
base of a rectangle lies along the x-axis while the upper two vertices lie
on the parabola
. Suppose that
the coordinates of the upper right vertex of the rectangle are (x,y). Find the
dimensions of the rectangle that would maximize the area.
- Which
point on the curve
is closest to the
point (1,0).
First Derivative Test
for Absolute Extreme Values
Suppose that c is a
critical number of a continuous function f defined on an interval.
(a) If
and 
, then f(c) is the absolute maximum value of f.
(b) If 
and 
, then f(c) is the absolute minimum value of f.