So I have the following question:
Find the extrema of the function
$$
f(x,y)=4x-6y
$$
Given the constraint
$$
4x^2-4x+9y^2-6y-2=0
$$
And determine whether these extrema are local/global on the constraint.
I found a max and min respectively at
$$
(\frac{1+\sqrt2}{2},\frac{1-\sqrt2}{3}) ,(\frac{1-\sqrt2}{2},\frac{1+\sqrt2}{3})
$$
with the value of f(x,y) at those points being
$$
4\sqrt2 ,-4\sqrt2
$$
I know these points a global maxima on the restriction/constraint, but I am having trouble proving that they are global.
Best Answer
Your constraint is an ellipsoid and your objective function is a straight line on 2-D. Therefore the straight line tangent to the ellipsoid should give you the global maximum/minimum.