Worked Examples of Extremum Problem

Example 1. Constant-sum, maximum-product principle. Given a positive number $S$. Prove that among all choices of positive numbers $x$ and $y$ with $x+y=S$, the product $xy$ is largest when $x=y=S.$

Proof. If $x+y=S,$ then $y=S-x$ and the product $xy$ is equal to $x(S-x)=xS-x^2.$ This quadratic polynomial has first derivative MATH which is positive for $x<\frac S2$ and negative for $x>\frac S2$. Hence the maximum of $xy$ occurs when $x=S/2,y=S-x=S/2.$ This can also be proved without the use of calculus. We simply write MATHand note that $f(x)$ is largest when $x=S/2.$

Example 2. Constant-product, minimum-sum principle. Given a positive number $P.$ Prove that among all choices of positive numbers $x$ and $y$ with $xy=P$, the sum $x+y$ is smallest when $x=y=\sqrt{P}$.

Proof. We must determine the minimum of the function $f(x)=x+P/x$ for $x>0$. The first derivative is MATH. This is negative for $x^2<P $ and positive for $x^2>P$ so $f(x)$ has its minimum at $x=\sqrt{P}$. Hence, the sum $x+y$ is smallest when $x=y=\sqrt{P}$.

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

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

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

Proof. Given $a>0,b>0$, let $P=ab$. Among all positive $x$ and $y$ with $xy=P $, the sum $x+y$ is smallest when $x=y=\sqrt{P}$. In other words, if $xy=P,$ then MATH In particular, MATH, so MATH. Equality occurs if and only if $a=b.$

Example 5. A block of weight $W$ is to be moved along a flat table by a force inclined at an angle $\theta $ with the line of motion, where MATH. 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 $\theta $ for which the propelling force needed to overcome friction will be as small as possible.

Solution. Let $F(\theta )$ denote the propelling force. It has an upward vertical component MATH so the net normal force pressing against the table is MATH The frictional force is $\mu N$, where $\mu $ is a constant called the coefficient of friction. The horizontal component of the propelling force is MATH. When this is equated to the frictional force, we get MATH from which we find
MATH
To minimize $F(\theta )$, we maximize the denominator MATH in the interval MATH At the endpoints, we have $g(0)=1$ and MATH. In the interior of the interval, we have
MATH
so $g$ has a critical point at $\theta =\alpha $ , where MATH This gives MATH. We can express $\cos \alpha $ in terms of $\mu $. Since MATH, we find MATH, so MATH. Thus MATH Since $g(\alpha )$ exceeds $g(0)$ and $g(\pi /2)$, the maximum of $g$ occurs at the critical point. Hence the minimum force required is
MATH

Example 6. Find the shortest distance from a given point $(0,b)$ on the $y$-axis to the parabola $x^2=4y$. (The number $b$ may have any real value.)

Solution. The quantity to be minimized is the distance
MATH
subject to the restriction $x^2=4y$. It is clear that when $b$ is negative the minimum distance is $\left| b\right| $. As the point $(0,b)$ moves upward along the positive $y$-axis, the minimum is $b$ until the point reaches a certain special position, above which the minimum is $<b.$ The exact location of this special position will now be determined.

First of all, we observe that the point $(x,y)$ that minimizes $d$ also minimizes $d^2$. (This observation enables us to avoid differentiation of square roots.) At this stage, we may express $d^2$ in terms of $x$ alone or else in terms of $y$ alone. We shall express $d^2$ in terms of $y.$

Therefore the function $f$ to be minimized is given by the formula
MATH

Although $f(y)$ is defined for all real $y$, the nature of the problem requires that we seek the minimum only among those $y\ge 0$. The derivative, given by MATH, is zero when $y=b-2$. When $b<2,$ this leads to a negative critical point $y$ which is excluded by the restriction $y\ge 0$. In other words, if $b<2$, the minimum does not occur at a critical point. In fact, when $b<2,$ we see that $f^{\prime }(y)>0$ when $y\ge 0$, and hence $f\,$ is strictly increasing for $y\ge 0$. Therefore the absolute minimum occurs at the endpoint $y=0$. The corresponding minimum $d$ is MATH

If $b\ge 2,$ there is a legitimate critical point at $y=b-2.$ Since MATH for all $y$, derivative $f^{\prime }$ is increasing, and hence the absolute minimum of $f$ occurs at this critical point. The minimum $d$ is MATH Thus we have shown that the minimum distance is $b$ if $b<2$ and is $2\sqrt{b-1}$ if $b\ge 2.$ (The value $b=2$ is the special value referred to above.)

Exercises

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

2. A farmer has $L$ 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 $A$ adjacent to a long stone wall. What dimensions require the least amount of fencing?

4. Given $S>0.$ Prove that among all positive numbers $x$ and $y$ with $x+y=S,$ the sum $x^2+y^2$ is smallest when $x=y.$

5. Given $R>0$. Prove that among all positive numbers $x$ and $y$ with $x^2+y^2=R,$ the sum $x+y$ is largest when $x=y$.

6. Each edge of a square has length $L$. Prove that among all squares inscribed in the given square, the one of minimum area has edges of length $L/\sqrt{2}.$

7. Each edge of a square has length $L$. 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 $R$. Find the radius $r$ and altitude $h$ of the right circular cylinder with largest lateral surface area $2\pi rh$ 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 $\sqrt{2}$ times that of the cylinder.

12. Given a right circular cone with radius $R$ and altitude $H$. 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 $R$ and altitude $H$.

14. Given a sphere of radius $R$. Compute, in terms of $R$, the radius $r$ and the altitude $h$ 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) $10$ and $10$; (b) $12$ and $18$.

18. If $a$ and $b$ are the legs of a right triangle whose hypotenuse is $1$, find the largest value of $2a+b$.

19. A truck is to be driven $300$ miles on a freeway at a constant speed of $x$ miles per hour. Speed laws require $30\le x\le 60.$ Assume that fuel costs $30$ cents per gallon and is consumed at the rate of $2+x^2/600$ gallons per hour. If the driver's wages are $D$ dollars per hour and if he obeys all speed laws, find the most economical speed and the cost of the trip if (a) $D=0$, (b) $D=1$, (c) $D=2$, (d) $D=3$, (e) $D=4$.

20. A cylinder is obtained by revolving a rectangle about the x-axis, the base of the rectangle lying on the $x$-axis and the entire rectangle lying in the region between the curve $y=x/(x2+1)$ and the $x$-axis. Find the maximum possible volume of the cylinder.

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