How Lagrange’s Mean Value Theorem is Applied in Real Life Situations

The role mean value theorem is extended by the Lagrange mean value theorem. According to the theorem, there exists a point on a curve between two points where the tangent is parallel to the secant line passing between these two points. The lone mean value theorem is another name for the Lagrange mean value theorem.

Let us learn more about the Lagrange mean value theorem, its proof, and its relationship with the Rolle mean value theorem through examples and frequently asked questions.

Definition:

According to the Lagrange mean value theorem, for each two points on a curve, there exists a point on the curve where the tangent drawn at this point is parallel to the secant through the two points.

Statement of Lagrange Mean Value Theorem:

Let f: [a, b]→ R be a continuous function on [a, b] and differentiable on [a, b] (a, b). The derivative of the function at point c is equal to the difference of the function values at these points, divided by the difference of the point values, if there is a point c in this interval (a, b).

                     f’(c)= f(b) – f(a)/b-a

          Graph of Lagrange mean value theorem

By graphing the equation as y = f, the Lagrange mean value theorem can be understood geometrically (x). The graph curve of y = f(x) passes through the points (a, f(a)), (b, f(b)), and a point (c, f(c)) is located halfway between these points and on the curve. The slope of the secant line running through (a, f(a)) and (b, f(b)) is  f(b) – f(a)/b-a . And f’ is the slope of the tangent line that touches the curve at point(c, f(c)) (c). The tangent at a point (c. f(c)) is parallel to the secant line going through the points (a, f(a)), (b, f(b)), and their slopes are equal, according to Lagrange’s mean value theorem. Hence we have  f’(c)= f(b) – f(a)/b-a.

Proof of Lagrange’s Mean Value Theorem:

Statement:  The Lagrange mean value theorem states that if a function f is continuous over the closed interval [a,b] and differentiable over the open interval (a,b), then there exists at least one point c in the interval (a,b) where the slope of the tangent at point c is equal to the slope of the secant through the curve’s endpoints, such that f’(c) = f(b) – f(a)/b-a .

Proof: Let g(x) be the secant line to f(x) that passes through (a, f(a)) and (b, f(b)). We know that the secant line has a slope of m=  f(b) – f(a)/b-a , The secant line is calculated using the formula y-y1=m (x- x1). The secant line also has the following equation.

y – f(a)= f(b) – f(a)/b-a  [x-a]

y = f(b) – f(a)/b-a  [x-a] + f(a)

Since the secant line’s equation is g(x) = y we have

g(x) = f(b) – f(a)/b-a  [x-a] + f(a)——–🡪 (1)

Let’s develop a function h(x) that equals the difference between the curve f(x) and the secant line g(x): h(x) = f(x) – g(x).

h(x) = f(x) – g(x)

The value of g(x) from the previous expression is used here.

h(x) = f(x) – [ f(b) – f(a)/b-a  [x-a] + f(a)]

Consider the function h(x), which is continuous on [a,b] and differentiable on [a,b] (a,b). Using Rolle’s theorem, there must be some x = c in (a,b) for h’(c) to equal 0.

h’(x) = f’(x) – f(b) – f(a)/b-a

For some c in (a,b), h’(c) = 0. Thus

h’(c) = f’(c) -f(b) – f(a)/b-a

f’(c) – f(b) – f(a)/b-a = 0

f’(c) = f(b) – f(a)/b-a

The Lagrange mean value theorem has therefore been established.

Application of Lagrange mean value theorem:

In the following areas, the Lagrange mean value theorem has been widely applied:

 (1) Prove the equation; 

(2) prove the inequality; 

(3) study the properties of derivatives and functions; and 

(4) prove the inequality. Prove the mean value theorem’s conclusion; 

(5) Determine the presence of the equation’s roots and their uniqueness. 

(6) To find the limit, use the mean value theorem.

Real Life Application of The Mean Value Theorem:

To check the accuracy of my speedometer, I employed The Mean Value Theorem. This information may be used to test the accuracy of our speedometer if I (my mother) set our car’s cruise control to 70 mph and timed how long it took us to travel one mile (mile marker to mile marker).

Importance of mean value theorem:

• Standard Deviation The importance of the theorem in mathematics is that it serves as the foundation for proving numerous theorems linked to integral and differential calculus; among these, we have the theorem that deals with the Proof of the First Derivative for Relative Extrema.
• The applicability of the Mean Value Theorem is based on the function’s continuity and differentiability, i.e., to utilize it in a specific situation, we must check these conditions. Its geometric meaning demonstrates that the tangent line is parallel to the secant at a point on the function’s graph.

 Conclusion;

The conclusion of the mean value theorem is that if a function f is continuous on the interval [a,b] and differentiable on the interval (a,b), then there exists a point “c” in the interval (a,b) where f′(c) equals the ratio of the difference of the functions f(a) and f(b).