Can be equal to zero, Lagrange multiplier is equal to zero. There is no constraint condition at this time, which is equivalent to directly deducing and calculating the extreme value. When the multiplier is not zero, there is a constraint at this time.

Lagrange mean value theorem, also known as Laplace theorem, is one of the basic theorems in differential calculus, which reflects the relationship between the overall average rate of change of a differentiable function in a closed interval and the local rate of change of a point in the interval. Lagrange mean value theorem is a generalization of Rolle mean value theorem and a special case of Cauchy mean value theorem. It is a weak form of Taylor formula (first-order expansion).

History:

1797, the French mathematician Lagrange put forward this theorem in the sixth chapter of his book Analytic Function Theory, and made a preliminary proof, so people named it Lagrange Mean Value Theorem. ?