Help understand Schwarz Inequality soqlution from Spivak's Calculus

Question

I'm given this inequality from Spivak's Calculus book:

And I need to do this:

Here is the answer from the answer book:

I am completely unsure what to do.I have $x_1, x_2, y_1, y_2$ in the inequality, but just $x$ and $y$ in the $(b)$ sub-problem.I fail to follow how the answer in the answerbook is derived.Could someone direct me?

Answer 1

In part (1) and (2) of the book's answer they plugged in $i=1$ and $i=2$ respectively for $x$ and $y$ as defined in part (b) (it looks like a typo in (2), the numerator should be $2x_2y_2$ on the left side of the inequality) .

This gives for $i=1$; $x=\frac{x_1}{\sqrt{x_1^2+x_2^2}}$ and $y=\frac{y_1}{\sqrt{y_1^2+y_2^2}}$, and for $i=2$; $x=\frac{x_2}{\sqrt{x_1^2+x_2^2}}$ and $y=\frac{y_2}{\sqrt{y_1^2+y_2^2}}$

The inequalities come from plugging the above $x$ and $y$ values into the inequality $2xy \le x^2+y^2$ for each value of $i$

Now you can add the inequalities and simplify to get the solution as given in the book.