How to tell of extrema lies on the boundary or interior of a function? (Lagrange Multiplier)

Question

For example:

Q: Find the extreme values of $f(x,y,z) = x + yz$ on the solid ellipsoid $x^2+2y^2+8z^2 \leq 32$.

The solution manual does:

" $f_x = 1$ not equal $0$, $f$ has no critical points.

-> all extrema must be on the boundary. "

But I don't understand why this means its not on the interior.

Thank you ..

Answer 1

Think of the $x,y$ plane. If the function that you are maximizing has no critical points on an interval, then the only other points that you can test for extrema are the end points of the interval. It's the same idea here. As $f_x=1$, there are no points of x for which there are extreme values, and so we only may evaluate the end points (boundary) of the surface for extreme values of $f$.

Answer 2

Every point of a set is precisely either an interior point or a boundary point (and interior/boundary is just a set of all interior/boundary points).

And if an inner point $x$ of a function is an extrema, then $f'(x) = 0$ (it has to be a critical point), since the set we bound the function on (in this case the ellipsoid) doesn't make any difference for such extrema and it will also be an extrema in $\mathbb{R}^n$. (I guess imagining it is enough - you are moving inside something, but if there's no critical point, the function is strictly monotone (considering it's differentiable) and you can't expect any extrema besides the set's boundary.)