I am preparing for an exam on Tuesday and I'm having problems with this excercise:
Decide, whether there are extremes for the function $f:\mathbb{R^3} \rightarrow \mathbb{R}, f(x,y,z) = xyz \hspace{2mm}$ on an elipsoid $3x^2 + 3y^2 + z^2 = 1 \hspace{2mm}$ If there are any, find them.
My solution so far:
$3x^2 + 3y^2 + z^2 -1 = 0$
$L(x,y,z, \lambda) = xyz - \lambda(3x^2 + 3y^2 + z^2 -1)$
Gradient:$\hspace{20mm}$Normal:
f'_x = yz \hspace{21mm} N'_x =6x
f'_y = xz \hspace{21mm} N'_y =6y
f'_z = xy \hspace{21mm} N'_z =2z
$(yz, xz, xy) = k(6x, 6y, 2z)$
$yz = 6kx$
$xz = 6ky$
$xy = 2kz$
$3x^2 + 3y^2 + z^2 = 1$
And I have a really hard time getting the points out of these four equations - I always run into a dead end. Can someone please give me a hint? Also I'm sorry for the formatting, but I haven't used LateX in a while.