I need to minimize and maximize $f(x,y,z)=xy^2z^3$, given that $x^2 + y^2 + z^2 = 6$.
According to the Lagrange multipliers calculator, there is an infinite number of points, where the function achieves the zero value.
But zero is neither minimum, nor maximum: $f(2,1,1)=2$, $f(-2,1,1)=-2$.
Why did the method fail to find the real maximum and minimum, and find the points, that are neither maximum, nor minimum?
Best Answer
It seems the online calculator fails, indeed from the given equation for the case $xyz \neq 0$ we find by elimination of $\lambda$
that is from the constraint
To check this result we can use the constraint to obtain the equivalent problem
$$g(x,z)=xz^3(6-x^2-z^2)$$
which indeed has maximum at $(x,z)=(\pm 1,\pm \sqrt 3)$ and minimum at $(x,z)=(\mp 1,\pm \sqrt 3)$.