Let the function f(x)f(x) be continuous in the closed interval [a, b][a, b] and differentiable in the open interval (a, b). Then a cc belongs to (a, b)(a, b), so: \frac{f(b)-f(a)}{b-a}=f'(c)b? af(b)？ f(a)=f .

Lagrange's mean value theorem is a basic theorem in calculus, which was named after Italian mathematician Joseph Louis Lagrange proved it in18th century. This theorem is widely used in mathematical analysis, differential equations, physics and many other fields. Lagrange's mean value theorem is expressed as follows: If a function f(x) is continuous in the closed interval [a, b] and derivable in the open interval (a, b), then at least one ξ belongs to (a, b).

The significance of this formula is that it gives the relationship between the average rate of change of a function in an interval and the difference between the function values at the end of the interval. If we know the function value at both ends of the interval, the length of the interval and the average change rate of the function in the interval (that is, f'(ξ) in the formula), we can calculate the function value at any point in the interval. The proof of Lagrange mean value theorem is based on Rolle theorem and Cauchy mean value theorem.

Lagrange mean value theorem is widely used. For example, in physics, we can use this theorem to solve the displacement, velocity or acceleration of an object in a certain time. In economics, we can use this theorem to solve the change of commodity price, demand or supply. In engineering, we can use this theorem to analyze and design complex systems such as control systems and circuit systems.

Although Lagrange mean value theorem is widely used, it also has some limitations. First of all, this theorem requires the function to be continuous in the closed interval, derivable in the open interval, and may not be satisfied in some special functions or special intervals. Secondly, this theorem can only give the relationship between the average change rate of a function in an interval and the difference of the function value at the end of the interval, but can't give the accurate value of the function at any point in the interval.