Example 1. *Constant-sum, maximum-product principle.* Given a positive
number
.
Prove that among all choices of positive numbers
and
with
,
the product
is largest when

Proof. If then and the product is equal to This quadratic polynomial has first derivative which is positive for and negative for . Hence the maximum of occurs when This can also be proved without the use of calculus. We simply write and note that is largest when

Example 2. *Constant-product, minimum-sum principle.* Given a positive
number
Prove that among all choices of positive numbers
and
with
,
the sum
is smallest when
.

Proof. We must determine the minimum of the function for . The first derivative is . This is negative for and positive for so has its minimum at . Hence, the sum is smallest when .

Example 3. Among all rectangles of given perimeter, the square has the largest area.

Proof. We use the result of Example 1. Let and , denote the sides of a general rectangle. If the perimeter is fixed, then is constant, so the area has its largest value when . Hence, the maximizing rectangle is a square.

Example 4. The geometric mean of two positive numbers does not exceed their arithmetic mean. That is, .

Proof. Given , let . Among all positive and with , the sum is smallest when . In other words, if then In particular, , so . Equality occurs if and only if

Example 5. A block of weight is to be moved along a flat table by a force inclined at an angle with the line of motion, where . Assume the motion is resisted by a frictional force which is proportional to the normal force with which the block presses perpendicularly against the surface of the table. Find the angle for which the propelling force needed to overcome friction will be as small as possible.

Solution. Let
denote the propelling force. It has an upward vertical component
so the net normal force pressing against the table is
The frictional force is
,
where
is a constant called the coefficient of friction. The horizontal component of
the propelling force is
.
When this is equated to the frictional force, we get
from which we find

To minimize
,
we maximize the denominator
in the interval
At the endpoints, we have
and
.
In the interior of the interval, we have

so
has a critical point at
, where
This gives
.
We can express
in terms of
.
Since
,
we find
,
so
.
Thus
Since
exceeds
and
,
the maximum of
occurs at the critical point. Hence the minimum force required is

Example 6. Find the shortest distance from a given point on the -axis to the parabola . (The number may have any real value.)

Solution. The quantity to be minimized is the distance

subject to the restriction
.
It is clear that when
is negative the minimum distance is
.
As the point
moves upward along the positive
-axis,
the minimum is
until the point reaches a certain special position, above which the minimum is
The exact location of this special position will now be determined.

First of all, we observe that the point that minimizes also minimizes . (This observation enables us to avoid differentiation of square roots.) At this stage, we may express in terms of alone or else in terms of alone. We shall express in terms of

Therefore the function
to be minimized is given by the formula

Although is defined for all real , the nature of the problem requires that we seek the minimum only among those . The derivative, given by , is zero when . When this leads to a negative critical point which is excluded by the restriction . In other words, if , the minimum does not occur at a critical point. In fact, when we see that when , and hence is strictly increasing for . Therefore the absolute minimum occurs at the endpoint . The corresponding minimum is

If there is a legitimate critical point at Since for all , derivative is increasing, and hence the absolute minimum of occurs at this critical point. The minimum is Thus we have shown that the minimum distance is if and is if (The value is the special value referred to above.)

1. Prove that among all rectangles of a given area, the square has the smallest perimeter.

2. A farmer has feet of fencing to enclose a rectangular pasture adjacent to a long stone wall. What dimensions give the maximum area of the pasture?

3. A farmer wishes to enclose a rectangular pasture of area adjacent to a long stone wall. What dimensions require the least amount of fencing?

4. Given Prove that among all positive numbers and with the sum is smallest when

5. Given . Prove that among all positive numbers and with the sum is largest when .

6. Each edge of a square has length . Prove that among all squares inscribed in the given square, the one of minimum area has edges of length

7. Each edge of a square has length . Find the size of the square of largest area that can be circumscribed about the given square.

8. Prove that among all rectangles that can be inscribed in a given circle, the square has the largest area.

9. Prove that among all rectangles of a given area, the square has the smallest circumscribed circle.

10. Given a sphere of radius . Find the radius and altitude of the right circular cylinder with largest lateral surface area that can be inscribed in the sphere.

11. Among all right circular cylinders of given lateral surface area, prove that the smallest circumscribed sphere has radius times that of the cylinder.

12. Given a right circular cone with radius and altitude . Find the radius and altitude of the right circular cylinder of largest lateral surface area that can be inscribed in the cone.

13. Find the dimensions of the right circular cylinder of maximum volume that can be inscribed in a right circular cone of radius and altitude .

14. Given a sphere of radius . Compute, in terms of , the radius and the altitude of the right circular cone of maximum volume that can be inscribed in this sphere.

15. Find the rectangle of largest area that can be inscribed in a semicircle, the lower base being on the diameter.

16. Find the trapezoid of largest area that can be inscribed in a semicircle, the lower base being on the diameter.

17. An open box is made from a rectangular piece of material by removing equal squares at each corner and turning up the sides. Find the dimensions of the box of largest volume that can be made in this manner if the material has sides (a) and ; (b) and .

18. If and are the legs of a right triangle whose hypotenuse is , find the largest value of .

19. A truck is to be driven miles on a freeway at a constant speed of miles per hour. Speed laws require Assume that fuel costs cents per gallon and is consumed at the rate of gallons per hour. If the driver's wages are dollars per hour and if he obeys all speed laws, find the most economical speed and the cost of the trip if (a) , (b) , (c) , (d) , (e) .

20. A cylinder is obtained by revolving a rectangle about the x-axis, the base of the rectangle lying on the -axis and the entire rectangle lying in the region between the curve and the -axis. Find the maximum possible volume of the cylinder.

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