# Thread: Spivak's Calculus - Schwarz Inequality

1. ## Spivak's Calculus - Schwarz Inequality

19. (a) Prove that if $\displaystyle x_1 = ny_1$ and $\displaystyle x_2 = ny_2$ for some number $\displaystyle n$, then equality holds in the Schwarz inequality. Prove the same thing if $\displaystyle y_1 = y_2 = 0$. Now suppose that $\displaystyle y_1$ and $\displaystyle y_2$ are not both $\displaystyle 0$, and that there is no number $\displaystyle n$ such that $\displaystyle x_1 = ny_1$ and $\displaystyle x_2 = ny_2$. Then

$\displaystyle 0 < (ny_1 - x_1)^2 + (ny_2 - x_2)^2$
$\displaystyle = n^2(y_1^2 + y_2^2) - 2n(x_1y_1 + x_2y_2) + (x_1^2 + y_2^2).$

Using Problem 18, complete the proof of the Schwarz inequality (i.e. prove that $\displaystyle x_1y_1 + x_2y_2 <= \sqrt{x_1^2 + x_2^2}\sqrt{y_1^2 + y_2^2}$).

I've proven the first two assumptions so far (by substitution), but I'm not sure how to prove the third. I know that I have to complete the square (which I've done in problem 18), but I can't seem to do that in this problem since there are double the variables. I've tried manipulating the inequality such that $\displaystyle x_1y_1 + x_2y_2 < \frac{1}{2n}(x_1^2 + y_2^2) + \frac{n}{2}(y_1^2 + y_2^2)$, but that doesn't seem to work either.

2. ## Re: Spivak's Calculus - Schwarz Inequality

Originally Posted by RogueDemon
19. (a) Prove that if $\displaystyle x_1 = ny_1$ and $\displaystyle x_2 = ny_2$ for some number $\displaystyle n$, then equality holds in the Schwarz inequality. Prove the same thing if $\displaystyle y_1 = y_2 = 0$. Now suppose that $\displaystyle y_1$ and $\displaystyle y_2$ are not both $\displaystyle 0$, and that there is no number $\displaystyle n$ such that $\displaystyle x_1 = ny_1$ and $\displaystyle x_2 = ny_2$. Then

$\displaystyle 0 < (ny_1 - x_1)^2 + (ny_2 - x_2)^2$
$\displaystyle = n^2(y_1^2 + y_2^2) - 2n(x_1y_1 + x_2y_2) + (x_1^2 + y_2^2).$

Using Problem 18, complete the proof of the Schwarz inequality (i.e. prove that $\displaystyle x_1y_1 + x_2y_2 <= \sqrt{x_1^2 + x_2^2}\sqrt{y_1^2 + y_2^2}$).

I've proven the first two assumptions so far (by substitution), but I'm not sure how to prove the third. I know that I have to complete the square (which I've done in problem 18), but I can't seem to do that in this problem since there are double the variables. I've tried manipulating the inequality such that $\displaystyle x_1y_1 + x_2y_2 < \frac{1}{2n}(x_1^2 + y_2^2) + \frac{n}{2}(y_1^2 + y_2^2)$, but that doesn't seem to work either.

3. ## Re: Spivak's Calculus - Schwarz Inequality

Do you know how to apply the discriminant to completing this proof? I can't figure it out. :/

4. ## Re: Spivak's Calculus - Schwarz Inequality

Originally Posted by RogueDemon
Do you know how to apply the discriminant to completing this proof? I can't figure it out. :/
$\displaystyle n^2(y_1^2 + y_2^2) - 2n(x_1y_1 + x_2y_2) + (x_1^2 + y_2^2)>0$

$\displaystyle n^2a - 2nb + c>0$ - Polynomial of degree 2.

$\displaystyle \Delta =b^2-4ac<0$(Why?)

5. ## Re: Spivak's Calculus - Schwarz Inequality

Originally Posted by Also sprach Zarathustra
$\displaystyle n^2(y_1^2 + y_2^2) - 2n(x_1y_1 + x_2y_2) + (x_1^2 + y_2^2)>0$

$\displaystyle n^2a - 2nb + c>0$ - Polynomial of degree 2.

$\displaystyle \Delta =b^2-4ac<0$(Why?)
By completing the square, it becomes evident that $\displaystyle n^2a - 2nb + c = n + \frac{-b + \sqrt{-(b^2 - ac)}}{a}$. Thus, it must be true that $\displaystyle b^2 - 4ac < 0$ since the contrary would imply that the expression is complex.

Though, how does this complete the proof of the Schwarz inequality?

6. ## Re: Spivak's Calculus - Schwarz Inequality

Wait, nvm, I got it. Thanks!