Find the maximum and minimum values of $f(x,y)=xy$ subject to the constraint $x^2−y=12$. Assume that $y≤0$ for this problem. Why is this assumption needed?
NOTE: I just want to know the necessity of the last part .why it should be $y\le0$.please be elaborate and yes,graphical analysis will be appreciated . Thank you.
Let's see what happens if this constraint is not asserted.
The problem then become to optimize $x(x^2-12)$ which is an cubic equation which has no maximum nor minimum, making the question too simple for you.