Rolle’s Theorem

Rolle’s Theorem is a specific example of Lagrange’s mean value theorem, which asserts that if a function f is defined in the closed interval [a, b] in such a way that it satisfies the following requirements, then the function f is said to satisfy the mean value theorem.

1)  The function f is continuous on the closed interval [a, b] and on the closed interval [a, b].

2) In addition, the function f differencing on the open interval is differentiable (a, b)

3) After all of this, let’s assume that there is at least one value of x that is between the two values of a and b, i.e. (a < c < b )  in such a way that f‘(c) = 0.

To put it another way, if a function is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a point x = c in the closed interval [a, b) where f'(c) = 0.

y = f(x) is continuous between x = a and x = b in the above graph, and at every point inside the interval, it is possible to draw a tangent to the curve, and ordinates that correspond to the abscissa and are equal, then there exists at least one tangent to the curve that is parallel to the x-axis.

When we look at this theorem from an algebraic perspective, it tells us that, given a polynomial function f (x) as its representation in x and two roots of the equation f(x)=0 as the values a and b, there exists at least one root of the equation f'(x) = 0 that lies between these values.

Yet, the opposite of Rolle’s theorem is not valid. Furthermore, there is a possibility that there are more than one value of x for which the theorem remains true; however, there is an extremely high probability of the existence of only one such value.

The Theorem of Rolle is as follows:

Rolle’s theorem can be expressed mathematically as follows:

Suppose f: [a, b] R is a continuous function on [a, b] and differentiable on (a, b), such that f(a) = f(b), where the real numbers a and b are used as examples. In that case, there exists at least one c in (a, b) such that f′(c) = zero.

Rolle’s Theorem Example

Example:

Check the validity of Rolle’s theorem for the function y = x2 + 2 for which a = –2 and b = 2.

Solution:

According to the statement of Rolle’s theorem, the function y = x2 + 2 is continuous in the interval [–2, 2] and differentiable in the interval (–2, 2).

From the given,

f(x) = x2 + 2

f(-2) = (-2)

2 + 2 = 4 + 2 = 6

f(2) = (2)2 + 2 = 4 + 2= 6

As a result, f(– 2) = f(2) = 6

As a result, the values of f(x) at –2 and 2 are the same.

As a result, f'(x) = 2x

In general, Rolle’s theorem states that there is at least one point c (– 2, 2) at which f′(c) = 0.

At c = 0, f′(c) = 2(0) = 0, where c = 0 (– 2, 2) and f′(c) = 2(0) = 0.

As a result, it was confirmed.

Conclusion

Rolle’s theorem is a specific version of the mean-value theorem in differential calculus. It is used in the field of analysis. According to Rolle’s theorem, if a function f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b) in such a way that f(a) = f(b), then f′(x) = 0 for some x with a range that includes b but is less than a. In other words, if a function f is continuous on the closed interval [a, b], then If a continuous curve goes through the same y-value (such as the x-axis) twice and has a unique tangent line (derivative) at every point of the interval, then somewhere in the middle of the curve, it has a tangent that is parallel to the x-axis. Another way to think about this is that if a curve goes through the same y-value twice, then it must be continuous. The theorem was given without a modern formal proof in the 12th century by the Indian mathematician Bhaskara II, but it wasn’t proven until 1691 by the French mathematician Michel Rolle. Rolle’s theorem is rarely applied since it can only establish that a solution exists; it does not establish the value of the solution; as a result, it is only useful in establishing that the mean-value theorem is true.