I was looking for example problems online and came across this problem:
a) Use Lagrange multipliers to find the absolute min and absolute max values of $f(x,y)=x^2+4y^2$ subject to the constraint $y=x^2-2$, if they exist
b) Sketch the level set diagram of $f(x,y)=x^2+4y^2$ and the constant curve $y=x^2-2$. Where are the candidate points that the Lagrange multipliers finds?
Solution:
For a) I get the points $\displaystyle{P=\left(\pm\sqrt{\frac{15}{8}},\frac{-1}{8}\right)}$. I can't real say if they are max or min points the function gives the same value at both points. I know the graph is a paraboloid and was trying to think of the constant curve projected onto the graph. I also tried plugging the constraint into the function and running optimization in one variable. The result is all critical values are complex.
How can we confirm if they are max, min or neither?
Part b) is easy enough to plot and think about level curves being tangent to the constraint curve. Here is a contour plot