A problem by Trevor B.

Let $f(x,y,z) = x^2 + y^2 +z^2$ and $g(x,y,z) = 6x-5y+3z=14$ . Taking Lagrange Multipliers gives:

$grad(f) = \lambda \cdot grad(g) \Rightarrow 2x = 6\lambda, 2y = -5\lambda, 2z = 3\lambda \Rightarrow y = -\frac{5x}{6}, z = \frac{x}{2}.$

or $6x -5(-\frac{5x}{6}) +3(\frac{x}{2}) = 14$ ;

or $x = \frac{6}{5}, y = -1, z = \frac{3}{5}$ .

The Hessian Matrix of $f, F(x,y,z) = 2 \cdot I_{3x3}$ , is positive-definite for all $x,y,z \in \mathbb{R}$ . Hence, the global minimum of $f$ computes to:

$f( \frac{6}{5}, -1, \frac{3}{5}) = \frac{36+25+9}{25} = \boxed{\frac{14}{5}}.$