Min & Max in 3-D

Lazy calculus Saturday here! Let $f(x,y,z) = x + 3y+2z, g(x,y,z) = x^2+y^2+z^2=2$ . Deployment of LaGrange Multipliers gives:

$grad(f) = \lambda \cdot grad(g) \Rightarrow 1 = \lambda(2x), 3=\lambda(2y), 2=\lambda(2z) \Rightarrow y =3x, z = 2x$

at which $g(x,3x,2x) = 14x^2 = 2 \Rightarrow x = \pm \frac{1}{\sqrt{7}}$ , and at which $a = 14(\frac{1}{\sqrt{7}}) = 2\sqrt{7}, b= 14(-\frac{1}{\sqrt{7}}) = -2\sqrt{7}$ . Hence, $-ab = \boxed{28}.$