Year 12 Proving homework question

Question

I am getting stuck on part a the following proving question. I don't know where to start for part b and I also have no idea how to link part b and part a together. Could someone please give me a hint on part b? Thank you so much for helping!

Consider the function $f(x) = \sum\limits_{k=1}^n (a_kx-1)^2$ where $a_1>0, a_2>0, ..., a_n>0$ are real.

a) Express $f(x)$ in the form $f(x) = Ax^2+Bx+C$ for real $A$, $B$, and $C$.

b) Show that $\sum\limits_{k=1}^n a_k^2 \ge \dfrac{1}{n}\left(\sum\limits_{k=1}^n a_k\right)^2.$

Answer 1

This kind of problem is an instance of the Cauchy-Schwarz Inequality:

$$ \left( \sum_{k = 1}^n a_kb_k \right)^2 \le \sum_{k = 1}^n a_k^2 \cdot \sum_{k = 1}^n b_k^2. \tag{$*$}$$

Notice that when $b_k = 1$ for all $k$ you get the inequality of part (b).

One way prove this inequality is to consider the function

$$ f(x) = \sum_{k = 1}^n (a_kx - b_k)^2. $$

If you write this as $Ax^2 + Bx + C$ then $A, B, C$ will be the three sums appearing in $(*)$. Then to get the inequality you note that $f(x) \ge 0$ for all $x$ so its discriminant, $B^2 - 4AC$ must be less than or equal to $0$. After handling some factors of $2$ you'll obtain $(*)$.

Answer 2

The second inequality follows directly from the identity:

$$n\sum\limits_{k=1}^n a_k^2=\left(\sum\limits_{k=1}^n a_k\right)^2+\sum\limits_{1\le i\lt j\le n}(a_i-a_j)^2$$