If the function f(x) satisfies: continuous on the closed interval [a, b]; It is differentiable in the open interval (a, b). Then there is at least one point ξ (a

The existence of f ′ (ξ) = (f (b)-f (a))/(b-a) or 0.

Make a secant between two points of the curve, the slope of the secant is (f(b)-f(a))/(b-a), and f'(c) is a tangent parallel to the secant, which is tangent to the curve at point C, so we should pay attention to the prerequisite of the mean value theorem. Although functions are continuous, they are not differentiable at x=c, and the mean value theorem requires all functions to be differentiable within the definition domain.

Lagrange mean value theorem;

Lagrange mean value theorem, also known as Lagrange theorem, is one of the basic theorems in differential calculus, which reflects the relationship between the overall average change rate of a differentiable function in a closed interval and the local change rate 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).

People's understanding of Lagrange's mean value theorem can be traced back to ancient Greece BC. When studying geometry, ancient Greek mathematicians came to the following conclusion: "The tangent line passing through the vertex of parabolic bow must be parallel to the bottom of parabolic bow". This is a special case of Lagrange theorem, and Archimedes, an ancient Greek mathematician, skillfully used this conclusion to find out the area of parabolic arch.

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.