1. Verify Theorem B when or , where , , are positive integers and .
[The first function vanishes for
and
. And
vanishes for
, which lies between
and
. In the second case we have to verify that the quadratic equation
has roots between
and
and between
and
.]
2. Show that the polynomials are positive when .
3. Show that is an increasing function throughout any interval of values of , and that increases as increases from to . For what values of is a steadily increasing or decreasing function of ?
4. Show that also increases from to , from to , and so on, and deduce that there is one and only one root of the equation in each of these intervals (cf. Ex. XVII. 4).
5. Deduce from Ex. 3 that if , from this that , and from this that . And, generally, prove that if and , then and are positive or negative according as is odd or even.
6. If and are continuous and have the same sign at every point of an interval , then this interval can include at most one root of either of the equations , .
7. The functions , and their derivatives , are continuous throughout a certain interval of values of , and never vanishes at any point of the interval. Show that between any two roots of lies one of , and conversely. Verify the theorem when , .
[If
does not vanish between two roots of
, say
and
, then the function
is continuous throughout the interval
and vanishes at its extremities. Hence
must vanish between
and
, which contradicts our hypothesis.]
8. Determine the maxima and minima (if any) of , , , , , . In each case sketch the form of the graph of the function.
[Consider the last function, for example. Here
, which vanishes for
,
, and
. It is easy to see that
gives a maximum and
a minimum, while
gives neither, as
is negative on both sides of
.]
9. Discuss the maxima and minima of the function , where and are any positive integers, considering the different cases which occur according as and are odd or even. Sketch the graph of the function.
10. Discuss similarly the function , distinguishing the different forms of the graph which correspond to different hypotheses as to the relative magnitudes of , , .
11. Show that has no maxima or minima, whatever values , , , may have. Draw a graph of the function.
12. Discuss the maxima and minima of the function when the denominator has complex roots.
[We may suppose
and
positive. The derivative vanishes if
This equation must have real roots. For if not the derivative would always have the same sign, and this is impossible, since
is continuous for all values of
, and
as
or
. It is easy to verify that the curve cuts the line
in one and only one point, and that it lies above this line for large positive values of
, and below it for large negative values, or
vice versa, according as
or
. Thus the algebraically greater root of (1) gives a maximum if
, a minimum in the contrary case.]
13. The maximum and minimum values themselves are the values of for which is a perfect square. [This is the condition that should touch the curve.]
14. In general the maxima and maxima of are among the values of obtained by expressing the condition that should have a pair of equal roots.
15. If has real roots then it is convenient to proceed as follows. We have where , . Writing further for and for , we obtain an equation of the form
This transformation from to amounts only to a shifting of the origin, keeping the axes parallel to themselves, a change of scale along each axis, and (if ) a reversal in direction of the axis of abscissae; and so a minimum of , considered as a function of , corresponds to a minimum of considered as a function of , and vice versa, and similarly for a maximum.
The derivative of with respect to vanishes if or if . Thus there are two roots of the derivative if and have the same sign, none if they have opposite signs. In the latter case the form of the graph of is as shown in Fig. 41a.
When and are positive the general form of the graph is as shown in Fig 41b, and it is easy to see that gives a maximum and a minimum.
In the particular case in which the function is and its graph is of the form shown in Fig. 41c.

The preceding discussion fails if , if . But in this case we have say, and gives the single value . On drawing a graph it becomes clear that this value gives a maximum or minimum according as is positive or negative. The graph shown in Fig. 42 corresponds to the former case.

[A full discussion of the general function
, by purely algebraical methods, will be found in Chrystal’s
Algebra, vol i, pp. 464–7.]
16. Show that assumes all real values as varies, if lies between and , and otherwise assumes all values except those included in an interval of length .
17. Show that can assume any real value if , and draw a graph of the function in this case.
18. Determine the function of the form which has turning values ( maxima or minima) and when and respectively, and has the value when .
19. The maximum and minimum of , where and are positive, are
20. The maximum value of is .
21. Discuss the maxima and minima of
[If the last function be denoted by
, it will be found that
22. Find the maxima and minima of . Verify the result by expressing the function in the form .
23. Find the maxima and minima of
24. Show that has no maxima or minima. Draw a graph of the function.
25. Show that the function has an infinity of minima equal to and of maxima equal to
26. The least value of is .
27. Show that cannot lie between and .
28. Show that, if the sum of the lengths of the hypothenuse and another side of a right-angled triangle is given, then the area of the triangle is a maximum when the angle between those sides is .
29. A line is drawn through a fixed point to meet the axes , in and . Show that the minimum values of , , and are respectively , , and .
30. A tangent to an ellipse meets the axes in and . Show that the least value of is equal to the sum of the semiaxes of the ellipse.
31. Find the lengths and directions of the axes of the conic
[The length
of the semi-diameter which makes an angle
with the axis of
is given by
The condition for a maximum or minimum value of
is
. Eliminating
between these two equations we find
32. The greatest value of , where and are positive and , is
33. The greatest value of , where and are positive and , is
[If
is a maximum then
. The relation between
and
gives
. Equate the two values of
.]
34. If and are acute angles connected by the relation , where , , are positive, then is a minimum when .