Let $a_1,\; ...\;, a_n$ and $b_1,\; ...\;, b_n\in \mathbb R$ be positive real numbers.
Find $$ max \;(a_1x_1 + a_2x_2 + ... + a_nx_n) $$ and $$ min \;(a_1x_1 + a_2x_2 + ... + a_nx_n) $$ over $x_1, \; ...\;, x_n\in \mathbb R$, $ $ subject to $b_1x_1^2 + b_2x_2^2 +\;...\;+b_nx_n^2 = 1$.
This question has already been asked, but I can't comment on it because I'm a new user... The answer provided there isn't satisfactory, as it uses a concept we haven't covered in class. This was given in the chapter about inner products, in my linear algebra class. Any thoughts?
An algebraic approach would be to introduce $y_i=\sqrt{b_i} x_i$, so that you are maximizing or minimizing $$\sum_{i=1}^n a_i b_i^{-1/2} y_i $$ under the constraint $\sum_{i=1}^n y_i^2 =1$. The Cauchy-Schwarz inequality $$\left|\sum_{i=1}^n a_i b_i^{-1/2} y_i\right| \le \sqrt{\sum_{i=1}^n a_i^2 b_i^{-1} } \sqrt{\sum_{i=1}^n y_i^2 } $$ provides both upper and lower bounds on $\sum_{i=1}^n a_i b_i^{-1/2} y_i$. To show they are attained, consider the equality case of the Cauchy-Schwarz inequality.