1. ## Schwarz's inequality

A) Given f(x) = ax^2 + 2bx + c with a>0. By considering the minimum, prove that f(x)>=0 for all real x if and only if b^2-ac<=0.

B) From A, let f(x)= (a1x+b1)^2 + (a2x+b2)^2 + ...... +(anx + bn)^2 and deduce Schwarz's inequality: (a1b1 + a2b2 + .... +anbn)^2 <= (a1^2 + a2^2 + ..... + an^2)(b1^2 + b2^2 + ..... + bn^2)

C) Show that equality holds in Schwarz's inequality only if there exists a real number x that makes aix equal to -bi for every value of i from 1 to n. (the i's are subs)

2. Originally Posted by mistykz
A) Given f(x) = ax^2 + 2bx + c with a>0. By considering the minimum, prove that f(x)>=0 for all real x if and only if b^2-ac<=0.

B) From A, let f(x)= (a1x+b1)^2 + (a2x+b2)^2 + ...... +(anx + bn)^2 and deduce Schwarz's inequality: (a1b1 + a2b2 + .... +anbn)^2 <= (a1^2 + a2^2 + ..... + an^2)(b1^2 + b2^2 + ..... + bn^2)
Part A is easy, you should know this fact.

Notice that $\displaystyle f(x)\geq 0$ for any $\displaystyle x\in \mathbb{R}$ because $\displaystyle (a_ix+b_i)^2 \geq 0 \mbox{ for }1\leq i\leq n$.
$\displaystyle f(x) = \left( \sum_{i=1}^n a_i^2 \right)x^2 + 2\left( \sum_{i=1}^n a_ib_i \right) x + \left( \sum_{i=1}^n b_i^2 \right)$.
But above if is necessary and sufficient for $\displaystyle b^2 - 4ac\geq 0$, thus:
$\displaystyle 4\left( \sum_{i=1}^n a_ib_i \right)^2-4\left( \sum_{i=1}^n a_i^2 \right)\left( \sum_{i=1}^n b_i^2 \right)\geq 0$.