Lagrange’s Mean value Theorem

Mathematical calculus relies heavily on Lagrange’s mean value theorem. Parmeshwara, an Indian mathematician from Kerala, initially proposed the mean value theorem in its original form in the 14th century. Rolle’s Theorem, a simpler version of this, was proposed by Rolle’s in the 17th century and proved solely for polynomials. Also, in the year 1823, Augustin Louis Cauchy proposed his version of the Mean Value Theorem that we use today. 

There is a point on a curve distance between two locations where the tangent is parallel to the secant running through those points. Using Lagrange’s mean value theorem, Rolle’s theorem was derived.

What is the Mean Value Theorem?

There must be at least one point on the curve f(x) where the tangent is parallel to the secant going through the two supplied points (a, f(a)),  (b, f(b)) according to the mean value theorem. If you have a function f(x) that is continuous and differentiable throughout an interval, then the mean value theorem can be applied here.  f(x) can be differentiated across the interval (a, b). Then there exists a c somewhere in (a, b). 

For example, f'(c) = f(b)-f(a)b-a

There is a tangent at c that follows the secant passing through the points (a, f(a)), (b, f(b)). For a statement to be proven, the mean value theorem is used. Rolle’s theory is also derived from the mean value theorem.

What Is the Lagrange Mean Value Theorem?

Lagrange Mean value theorem asserts that for every two locations on a curve, there exists a point where the tangent drawn at this point is parallel to the secant through these places.

Statement of Lagrange Mean Value Theorem: Functions f can be defined so that f: [a, b], R is a continuous function on [a, b] and differentiable on (a, b). The derivative of the function at point c is equal to the difference between the function values at these locations, divided by the difference between the point values if a point c exists in the interval (a, b).

f'(c) = f(b)-f(a)b-a

What is Rolle’s Theorem?

This theorem in calculus argues that if the slope of the tangent line to the graph of a differentiable (real-valued) function is zero, then the function must have at least one fixed point between the two points in question. It is named after Michel Rolle, a French mathematician who first proved it. It is a specific case of the mean value theorem. It is also known as Rolle’s mean value theorem, which is also known as the first mean value theorem. 

Lagrange Mean Value Theorem vs Rolle’s Mean Value Theorem

The Lagrange mean value theorem is an extension of Rolle’s mean value theorem that has been published in the literature. Rolle’s mean value theorem defines a function y = f(x), such that the function f : [a, b] → R be continuous on [a, b] and differentiable on (a, b). Here we have the function values equal such that f(a) = f(b), where a and b are some real numbers. Then there exists some c in (a, b) such that f′(c) = 0. Geometrically for the graph of the function y = f(x) the line joining (a, f(a)), (b, f(b)) is parallel to the x-axis, and the slope of the tangent at the point (c, f(c))), is equal to zero, f'(c) = 0. The tangent line at point (c, f(c)) is parallel to the x-axis and has a slope of zero according to geometric considerations (i.e. the function y = f(x)).

It has also been defined for a function y = f(x) where the rolle’s mean value theorem is applied to the function f(x) (a, b). Hence, the difference between these points’ functions and their respective point values, c in this interval, is equal to the derivative (a, b).

Conclusion

The mean value theorem which includes Lagrange’s mean value theorem, and rolle’s theorem is an integral part of differential calculus. From this article, we have tried to learn about Lagrange mean value theorem and how it’s being used. The Lagrange mean-value theorem has been frequently applied in the following areas: Derivative and function qualities must be studied in order to prove or disprove an equation or an inequality. The mean value theorem’s conclusion must be proved.  Find the equation’s roots and verify their existence and uniqueness.