# Mathematics | Lagrange’s Mean Value Theorem

**S**uppose

be a function satisfying these conditions:

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 path, i.e.,

**Example:** Verify mean value theorem for f(x) = x^{2} 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.

f(2) = 4

and f(4) = 16

Mean value theorem states that there is a point c ∈ (2, 4) such that

But

which implies c = 3. Thus at c = 3 ∈ (2, 4), we have

This article has been contributed by Saurabh Sharma.

If you would like to contribute, please email us your interest at review-team@geeksforgeeks.org

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above

Attention reader! Don’t stop learning now. Practice GATE exam well before the actual exam with the subject-wise and overall quizzes available in **GATE Test Series Course**.

Learn all **GATE CS concepts with Free Live Classes** on our youtube channel.