Using the Lagrange multipliers method I have to find the absolute maximum and minimum value of $f(x, y)=x^2+y^2-x-y+1$ in the unit disc.
So, I have to find the extremas of $f(x, y)=x^2+y^2-x-y+1$ subject to $x^2+y^2 \leq 1$, or not??
Do we not apply Lagrange multipliers method when we have a function $f(x,y)$ and a constaint $g(x, y)=0$??
So, shouldn't we have to have an equality at the constraint??
But in this case we have an inequality… What do we do??
Best Answer
It is a closed region, so max and min must occur. They can only occur on the boundary or at critical points of the function. So you can use the following steps:
Step 1: Find all the critical points of the function, and check whether they are in the constraint region.
Step 2: Use regular Lagrange multiplier method on the boundary of the disk.
Then combine the results from the two steps to find the max and min.