I'm given the function $$f(x,y,z) = xy+yz$$ with the constraints \begin{align*} x+y+z=1\quad \wedge \quad x^2+y^2+z^2=6 \end{align*} I've tried to determine the extreme values with lagrange multipliers and through parametrisation.
I'm pretty confident with my solution with lagrange multipliers since it's also the solution Wolfram Alpha computes (see here)
But when I do parametrisation, I get that \begin{align*} &f(x,y,z) = y(x+z) \quad \wedge \quad x+z = 1-y\\ \Rightarrow \quad &f(x,y,z) = y-y^2 \end{align*} and with the last constraint I get that $-\sqrt{6}\leq y\leq \sqrt{6}$.
The problem is that the maximum value of the two methods matches but the minimum value doesn't.
It's at $y = 1/3 - \sqrt{34}/3$ for lagrange mult. and at $y=-\sqrt{6}$ for parametrisation.
Am I doing something wrong?