Lagrange multipliers problem

Question

I have a two variables function: $f(x,y)=3x+y$ and I wish to find its minimum and maximum values with the constraint $\sqrt{x} +\sqrt{y} =4$. According to the answer, there is a minimum and a maximum. However, I found only 1 point, the minimum, and couldn't find another. MAPLE also found only the minimum. The answer in the answers sheet say that 16 is between the min and max, I can't see why if there is only 1 point. Or maybe there are more ?

Answer 1

As has been remarked in the comments, the Lagrange-multiplier method only looks for level lines, planes, surfaces etc. which are tangent to the "constraint" curve, surface, etc. The constraint $ \ \sqrt{x} \ + \ \sqrt{y} \ = \ 4 \ $ is the portion of the astroid curve (marked in blue) in the first quadrant; "Lagrange" gives you the level curve $ \ f(x) \ = \ 3x \ + \ y \ = \ c \ $ which is tangent to the constraint curve. Since the level curve is a straight line and the "quarter-astroid" is concave upward, it can only intersect the constraint curve at one point (on the red line). So the minimum value for $ \ f ( x ) \ $ is thereby located.

The maximum must be found (if it exists) in the same way that we look for extrema, for instance, for a function of one variable in a closed interval: after locating any local extrema in the "interior" of the interval, we must also check the endpoints of the interval. The quarter-astroid has an $ \ x-$ and a $ \ y-$ intercept, so the values of $ \ f(x) \ $ must also be checked at each of those points. (The green line is the "level curve" for $ \ 3x \ + \ y \ = \ c' \ $ .) [I won't spoil things by giving the extremal values themselves.]