How to judge minimax by Lagrange multiplier method

When using Lagrange multiplier method, the method to judge the maximum or minimum usually includes observing the change of the first derivative of the function on both sides of the stagnation point and checking the second derivative of the function. The following is a detailed introduction:

1. Observe the change of the first derivative of the function on both sides of the stagnation point. If one side of the first derivative is positive (increasing) and the other side is negative (decreasing), the stagnation point is minimal; On the contrary, it is the maximum.

2. Check the second derivative of the function. If the second derivative at the stagnation point is greater than zero, it means that the point is a minimum; If the second derivative is less than zero, this point is the maximum.

3. In addition, for continuous functions on bounded closed sets, we can know from the maximum theorem that if there is a maximum or a minimum, then at the stagnation point of Lagrange multiplier method, the maximum is the maximum and the minimum is the minimum.