Unknown

226 7. Approximations and Inequalities

12. Determine u so that the zeros of the polynomial t2 - (3u + 1)t +
    (2u2 - 3u - 2) are real and the sum of their squares is minimal.


13. Let m and n be positive integers and let 2, y be positive reals. Show
    that
    (z)“‘” 2 (f)* (I)“.


14. Determine the largest value of y such that

for all x > 0.

15. Let f(t) be an irreducible polynomial of degree n exceeding 1 over
    Z, and suppose that r is a zero of f(t). Show that there is some
    constant k which depends on f and r such that Ip/q - rl > k/q” for
    each rational p/q written in lowest terms with q a positive integer.


16. Find the maximum and minimum values of
    x+1 z+l
    xy+x+l

+ Y+l +

yz+y+l *x+*+1

subject to the conditions that x, y, z 2 0 and xyz = 1.

7.5 Other Problems

1. Let n be a positive integer greater than 2 and let f be any polyno-
   mial of degree not exceeding n - 2. If al, or,... , a, are any complex
   numbers and p(t) = (t - al)(t - ~2)... (t - a,), prove that

n

c

f(Ui) _ 0.

izl P’(%)

2. Let a, b, c, d be distinct complex numbers. Show that

(4

a4 b4 C4

(a - b)(a - c)(u - d) + (b - u)(b - c)(b - d) + (c - u)(c - b)(c - d)

+ (d - a)(df b)(d - c)

= u+b+c+d.