# Proving an Inequality

1. Feb 2, 2005

Hello all:

Show that if $$a > 0$$, $$ax^2 + 2bx + c \geq 0$$ for all values of x if and only if $$b^2 - ac \leq 0$$. Ok so I rewrote $$ax^2 + 2bx + c$$ as $$a(x+ \frac{b}{a})^2 + \frac{ac-b^2}{a}$$ Now how would I work with this expression?

Also if you are given $$(a_1x + b_1)^2 + (a_2x + b_2)^2 + ... + (a_nx + b_n)^2$$ how would you prove Schwarz's Inequaliity? Would it be:

Schwarz's Inequality

$$(a_1b_1 + a_2b_2 + ... + a_nb_n)^2 \leq (a_1^2 + ... + a_n^2)(b_1^2+...+b_n^2)$$

So $$(a_1x^2 + 2a_1xb_1 + b_1^2) + (a_2x^2 + 2a_2x + b_2^2) + (a_nx^2 + 2a_nxb_n + b_n^2)$$. So factoring we have $$x^2(a_1+a_2+ ... + a_n) + 2x(a_1b_1 + a_2b_2 + ... + a_nb_n) + (b_1^2 + b_2^2 + ... + b_n^2)$$ Now how would I prove Schwarz's inequality from here?

Thanks a lot

2. Feb 2, 2005

### Gokul43201

Staff Emeritus
$$a(x+ \frac{b}{a})^2 + \frac{ac-b^2}{a} \geq 0 ~ for~all~x$$

Theifirst term is clearly a positive number but will be zero only when x = -b/a. So, if the second term were negative, the sum would be negaitve for some values of x (in particular, for x = -b/a). This is not allowed. Hence, the second term must be ...

3. Feb 2, 2005

the second term must be positive or this implies that $$b^2 - ac \leq 0$$

4. Feb 2, 2005

is my approach to the second question correct?

Thanks

5. Feb 2, 2005

### Gokul43201

Staff Emeritus
The question requires you to prove the converse too ("if, and only if"). But this is just working backwards along the same steps, and is trivial to do.

Looking at #2 now....

6. Feb 2, 2005

### Gokul43201

Staff Emeritus
Yor approach here is correct. You seem to have made one small error, though.

Starting from $$(a_1x + b_1)^2 + (a_2x + b_2)^2 + ... + (a_nx + b_n)^2 \geq 0$$

you should get

$$x^2(a_1^2+a_2^2+ ... + a_n^2) + 2x(a_1b_1 + a_2b_2 + ... + a_nb_n) + (b_1^2 + b_2^2 + ... + b_n^2) \geq 0$$

Now use the result you proved in #1 (since the coefficient of the x^2 term is positive), and you are home.

7. Feb 2, 2005

I just used the fact that $$b^2 - ac \leq 0$$