Lagrange’s Mean Value Theorem

1) f(x) is continuous in the closed interval a ≤ x ≤ b

2) f(x) is differentiable in the open interval a < x < b

Then according to Lagrange’s Theorem, there exists at least one point ‘c’ in the open interval (a, b) such that:

We can visualize Lagrange’s Theorem by the following figure

In simple words, Lagrange’s theorem says that if there is a path between two points A(a, f(a)) and B(b, f(a)) in a 2-D plain then there will be at least one point ‘c’ on the path such that the slope of the tangent at point ‘c’, i.e., (f ‘ (c)) is equal to the average slope of the 

Example: Verify mean value theorm for f(x) = x² in interval [2,4].

Solution: First check if the function is continuous in the given closed interval, the answer is Yes. Then check for differentiability in the open interval (2,4), Yes it is differentiable.