This is an excerpt from my math models textbook. It's about Lagrange Polynomials which is a technique that lets you fit a polynomial to a set of any number of unique points (x_1,y_1) ... (x_n,y_n) so long as all your x-values are different (otherwise it wouldn't be a function, and couldn't be a polynomial). The polynomial you'll calculate will be the unique, lowest degree polynomial that passes through all points.
I'm not sure sure what the stated premise is supposed to even mean.
Replace the words "Convince yourself" with "You can verify" and it might make more sense.
No, I got that part, but I don't think I understand the significance of the indexed y values and their relationships to the indexed x values. The criterion seems to suggest that P3(xn)=yn for each, but that strikes me as something that is defined as a constraint rather than something that is to be proved. Also, I woke up then and now so that might be playing a factor in my confusion.
OK, you got it then, I believe. P3 is specifically built so that P3(xn)=yn for n from 1 to 4. The proof lies in its construction. I guess the sentence can be understood as "we know it works because we built it like that, however you may verify it yourself"
I feel like the sentence also means "it's kinda obvious when you think about it, so we won't explain, but it's actually important, so you probably should make sure you agree".
The function should be cubic, so you should be able to write it in the form "f(x) = ax^3 + bx^2 + cx + d". You could work out the entire thing to put it in that form, but you don't need to.
Since there are no weird operations, roots, divisions by x, or anything like that, you can just count how many times x might get multiplied with itself. At the top of each division, there are 3 terms with x, so you can quite easily see that the maximum will be x^3.
It's useful to know what the values x_i and x_y are though. They describe the 3 points through which the function should go: (x_1, y_1) to (x_3, y_3).
That also makes the second part of the statement ready to check. Take (x_1, y_1) for example. You want to be sure that f(x_1) = y_1. If you replace all of the "x" in the formula by x_1, you'll see that everything starts cancelling each other out. Eventually you'll get "1 * y_1 + 0 * y_2 + 0 * y_3", thus f(x_1) is indeed y_1.
They could have explained this a bit better in the book, it also took me a little while to figure it out.