(2) can be derived in (a, b)

Then there is at least one point f' (c) = [f (b)-f (a)]/(b-a) a.

Prove that the original function f(x)={[f(b)-f(a)]

Let the auxiliary function g (x) = f (x)-{[f (b)-f (a)]/(b-a)} x.

It is easy to prove that this function satisfies the condition in this interval:

1.G(a)= G(b)；

2.G(x) is continuous in [a, b];

3.G(x) is differentiable in (a, b).

This is the condition of Rolle's theorem, which is proved by Rolle's theorem.

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Extended data

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).

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.

References:

Baidu encyclopedia-Lagrange mean value theorem