Help with Constrained optimization, and Lagrange Multipliers

Question

How can I find the extremal values of the function $f(x,y,z)=5x-2y+z+17$ under the constraint $x^2 + y^2 + z^2=30$?

Answer 1

consider the function $$f(x,y,z,\lambda)=5x-2y+z+17+\lambda(x^2+y^2+z^2-30)$$ and solve the system $$f_x=0$$ $$f_y=0$$ $$f_z=0$$ $$f_{\lambda}=0$$

Answer 2

Note that you are extremising a linear function on the surface of the sphere.

This is of the form extremise $c^T x$ subject to $\|x\|^2 = 30$.

We know from Cauchy Schwarz that $|c^T x| \le \|c\| \|x\|$ and that $|c^T x| = \|c\| \|x\|$ iff $x $ is a multiple of $c$.

Hence we look for solutions $x = \pm \sqrt{30} {1 \over \|c\|} c$.