212 
DERIVATIVES AND INTEGRALS 
[VI 
24. Discuss {x 2 -x + l)j{x 2 + x+l), {x 2 — x+3)/{x 2 — 3x + 2). 
25. The maximum and minimum of {x+a){x + b)/{x-a){x — b), where a 
and b are positive, are 
fy/a + Jb\ 2 
[Ja-JbV 
\Qa-QbJ ’ 
W a + s/b) 
26. The maximum value of {x — l) 2 /{x+1) 3 is ^ r . 
27. Discuss the maxima and minima of 
x (x—l)l{x 2 +3x+S), x*/{x—\){x — 3) 3 . 
28. Discuss the maxima and minima of 
{x - 1 ) 2 (3a’ 2 - 2x- 37)/{x + 5) 2 (3a 2 - 14a - 1). 
{Math. Trip. 1898.) 
[If the function be denoted by P{x)jQ{x), it will be found that 
P'Q - PQ'=72 (a - 7) (a - 3) (a -1) (a+1) (a+2) (a+5).] 
29. Find the maxima and minima of a cos x + h sin a. Verify the result 
by expressing the function in the form A cos (a — a). 
30. Find the maxima and minima of 
a 2 cos 2 a+b 2 sin 2 a, A cos 2 a + 2H cos a sin a + B sin 2 a. 
31. Show that sin (a + a)/sin (a + 6) has no maxima or minima. Draw a 
graph of the function. 
32. The least value of a 2 sec 2 a + b 2 cosec 2 a is {a + b) 2 . 
33. Show that tan 3a cot 2a cannot lie between | and |. 
34. Find the maxima and minima of (1 + 2a arc tan a)/(l +a 2 ). 
35. The base of a triangle is equal to a, and the ratio of the other two 
sides is r. Show that the maximum value of its area is \a 2 r\{r 2 ~ 1). 
36. A line is drawn through a fixed point {a, b) to meet the axes OX, OY 
in P and Q. Show that the minimum values of PQ, OP+OQ, and OP. OQ 
2 2 3 
are respectively (or+ ZF) 5 , {Ja + Jb) 2 , Aab. 
37. A tangent to an ellipse meets the axes in P and Q. Show that the 
least value of PQ is equal to the sum of the semiaxes of the ellipse. 
38. Find the lengths and directions of the axes of the conic 
ax 2 + 2 hxy + by 2 = 1. 
[The length r of the semidiameter which makes an angle 6 with the axes 
of x is given by 
1 jr 2 = a cos 2 6 + 2h cos 6 sin 6 + b sin 2 6, 
The condition for a maximum or minimum value of r is tan 26 = 2h/{a - b): 
eliminating 6 between these two equations we find 
{a-(l/r 2 )}{6-(l/r 2 )}=A 2 ] 
39. The greatest value of the product of two positive numbers whose 
sum is constant is obtained by supposing them equal. Extend the result to 
any number of numbers.