Suppose you want to minimize $\displaystyle a_{1}^{2} + \cdots + a_{n}^{n} $ subject to the constraints $\displaystyle a_i >0 $ and $\displaystyle a_1+ \cdots + a_n = 1 $.

So in the Cauchy Schwarz Inequality do the following: Take $\displaystyle b_1 = \cdots = b_n = 1 $. So we have:

$\displaystyle (a_{1}^{2}+ \cdots + a_{n}^{2})(b_{1}^{2} + \cdots b_{n}^{2}) \geq (a_{1}b_{1} + \cdots +a_{n}b_{n})^2 $.

$\displaystyle a_{1}^{2} + \cdots + a_{n}^{2} \geq (a_1+ \cdots + a_n)^2 $.

$\displaystyle a_{1}^{2} + \cdots + a_{n}^2 \geq 1 $

So $\displaystyle a_1 = \cdots = a_n = \frac{1}{\sqrt{n}} $?