<s>[INST] Min-Max points with lagrange multipliers "Find the min and max points of the function
$$f(x) = x^ 2 +yz-5$$
on the sphere
$$x^2+y^2+z^2=1$$
using Lagrange multipliers".
I'm having problem with equations while finding the points:
$$\nabla f(x) = \nabla g(x) \lambda$$
$$ 2x = 2x \lambda $$
$$z = 2y \lambda $$
$$y = 2z \lambda $$
$$ x^2+y^2+z^2-1 = 0 $$
 [/INST] Then, if $\lambda=0$, then $(x,y,z)=(0,0,0)$ but since  $x^2+y^2+z^2-1=0$ we have a contradiction. Thus, consider $\lambda\not=0$ and $x\not=0$, then since $2x=2x\lambda$ then $\lambda=1$ and then $(x,y,z,\lambda)=(\pm 1,0,0,0)$ because $z=2y=2(2z)$ give $z=0$ but also $y=2z=2(2y)$ give $y=0$. However, for $\lambda \not=0,1$ and $x=0$ also we have $z=2y\lambda=2(2z)\lambda^2 $ give $\lambda=\pm \frac{1}{2}$. If $\lambda =-\frac{1}{2}$ we have $(x,y,z,\lambda)=(0,\pm \frac{1}{\sqrt{2}}, \mp \frac{1}{\sqrt{2}}, -\frac{1}{2})$. If $\lambda=\frac{1}{2}$ we have $(x,y,z,\lambda)=(0,\mp\frac{1}{\sqrt{2}},\mp\frac{1}{\sqrt{2}},\frac{1}{2})$. Thus, critical points are given by
$\boxed{(\pm 1,0,0)}$ and $\boxed{(0,\pm\frac{1}{\sqrt{2}},\mp\frac{1}{\sqrt{2}})}$. Then, we need check it into $f(x,y,z)=x^2+yz-5$ in order to find the extremes values.
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