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.