Consider the function $$f(x, y)=x^2 y -2xy^2 +4.$$ Find the absolute maxima and minima of $f$ constrained to $2x-4y+1=0$ with $x\in [-1, 1]$.
My solution: the Weierstrass theorem guarantees that the function has absolute minima and maxima. The Lagrange multipliers method has allowed me to find only the point $(-1/4, 1/8)$. My question is: how to understand if it is a maximum or a minimum? When I have two or more points, I evaluate the function in those points and the greatest value corrispond to the maximum point and the lowe to the minimum point.
In this case, how to proceed? Thank you in advance!
EDIT: Plotted function.
Here's the surface and the constraint line:
Although the curvature on the constraint line is low, there is a single maximum, as is to be expected in a quadratic. The minima occur at the boundaries of the region... and Lagrangian methods do not find those.
If you're given a bounded region, you may have to check boundaries to find maxima and minima. After all, consider the trivial problem of finding the maxima and minima of $f(x)=x$ in the range $x \in [0,1]$. Even basic calculus will not yield the extrema. You must check boundaries. The same sometimes occurs with Lagrangian methods.