5.5.35 If you have studied vector dot-product, you
should be able to give a geometric interpretation of the
Cauchy-Schwarz inequality. Think about magnitudes,
cosines, etc.
5.5.36 Here's another way to prove Cauchy-Schwarz
that employs several useful ideas: Define
f(t) := (alt+bd+(a 2 t+bd +···+(ant+bn)^2.
Observe that f is a quadratic polynomial in t. It is pos
sible that f(t) has zeros, but only if al/bl = a 2 /b 2 =
. .. = anlbn. Otherwise, f(t) is strictly positive. Now
use the quadratic formula, and look at the discriminant.
It must be negative; why? Show that the negativity of
the discriminant implies Cauchy-Schwarz.
5.5.37 Give a quick Cauchy-Schwarz proof of the in
equality in Example 5.5.18.
5.5.38 Let ai, a 2 , ... , an be positive, with a sum of 1.
n
Show that � aT � lin.
1 =1
5.5.39 If a, b, c > 0, prove that
(a^2 b + b^2 c + c^2 a)(ab^2 + bc^2 + ca^2 ) � 9a^2 b^2 c^2.
5.5.40 Let a,b,c � O. Prove that
5.5.41 Let a,b,c, d � O. Prove that
I I 4 16 64
- abc +-+-+-d > -a+b+c+d.
5.5 INEQUALITIES 1875.5.42 (USAMO 1983) Prove that the zeros of
x^5 + ax^4 + bx^3 + cx^2 + dx + e = 0
cannot all be real if 2a^2 < 5b.
5.5.43 Let x, y, z > 0 with xyz = 1. Prove that
x+y+z S � +i +z^2.
5.5.44 Let x,y, z � 0 with xyz = 1. Find the minimum
value of
�+_y_+_z_.
y+z x+z x+y
5.5.45 (IMO 1984) Prove that
OS yz+zx+xy-2xyz S 7/27,
where x, y and z are non-negative real numbers for
which x + y + z = 1.
5.5.46 (Putnam 1968) Determine all polynomials
with all coefficients equal to 1 or -I that have only
real roots.
5.5.47 Let al,a 2 ,... ,an be a sequence of positive
numbers, and let bl , b 2 , ... , bn be any permutation of
the first sequence. Show that� + a (^2) + ... + an > n.
bl b 2 bn -
5.5.4 8 Let al ,a 2 ,... ,an and bl ,b 2 ,. .. ,bn be increas
ing sequences of real numbers and let XI ,X 2 , ..• ,Xn be
any permutation of bl ,b 2 , ... , bn. Show that
�aibi � �aixi.