🧵 How does the discriminant based proof for C-S inequality downs on your mind ?

Anonymous at Tue, 23 Apr 2024 16:57:50 UTC No. 16142299

years ago our analysis teacher introduced this question on the problem set for first chapter of the course:

[math]\left| \sum_{i=1}^{n}x_iy_i \right|\le \sqrt{\sum_{i=1}^{n}x_i^2}\sqrt{\sum_{i=1}^{n}y_i^2}[/math] .... Cauchy-Shwartz inequality
for the solution he adopted:
consider the expression
[math]p(t)=\sum_{i=1}^{n}(tx_i+y_i)^2[/math]
[math]p(t)=\left( \sum_{i=1}^{n}x_i^2 \right)t^2+\left( 2\sum_{i=1}^{n}x_iy_i \right)t+\left( \sum_{i=1}^{n}y_i^2 \right)[/math]
Which is just a quadratic polynomial for variable t. since [math]p(t)\ge 0[/math] for all t than the discriminant [math]\Delta[/math] must be a negative number (if it was positive than the polynomial will change sign twice).
[math]\Delta\le 0[/math]
[math]4\left( \sum_{i=1}^{n}x_iy_i \right)^2-4\left( \sum_{i=1}^{n}x_i^2 \right)\left( \sum_{i=1}^{n}y_i \right)\le 0[/math]
thus the C-S inequality. Now I remember when I and a collegue saw that problem, he said "This is an extraordinary formula that two great mathematicians put together their minds to prove",
but than I read in a french book that Cauchy proved this version all by himself, Schwatz proved another one with double integrals, and the generalized case was done by another mathematician.
But what boggles my mind is how any human would think of such a proof?