TITLE: How to know whether Lagrange multipliers gives maximum or minimum?
QUESTION [36 upvotes]: My book tells me that of the solutions to the Lagrange system, the smallest is the minimum of the function given the constraint and the largest is the maximum given that one actually exists.
But what if we only have one point as a solution? How to know whether Lagrange multipliers gives maximum or minimum?

REPLY [6 votes]: On a closed bounded region a continuous function achieves a maximum and minimum. If you use Lagrange multipliers on a sufficiently smooth function and find only one critical point, then your function is constant because the theory of Lagrange multipliers tells you that the largest value at a critical point is the max of your function, and the smallest value at a critical point is the min of your function. Thus max = min, i.e. the function is constant. Also note that "critical point" should probably be called something else, like "point of interest" because usually critical points are defined as points where the gradient is zero.