Minimum of $x^2+y^2+z^2$ subject to $ax+by+cz=1$ 
If $ax+by+cz=1$, what is the minimum of $x^2+y^2+z^2$?

It is obvious that we can do Lagrangian multiplier
$$W=x^2+y^2+z^2-\lambda (ax+by+cz-1)$$

Using the Cauchy-Schwartz Inequality:
$(x^2+y^2+z^2)\cdot (a^2+b^2+c^2)\geq (ax+by+cz)^2$ , Now Given $(ax+by+cz) = 1$
So $\displaystyle (x^2+y^2+z^2)\geq \frac{1}{(a^2+b^2+c^2)}$ and equality hold when $\displaystyle \frac{x}{a} = \frac{y}{b} = \frac{z}{c}.$
