Introduction to Rolle’s theorem

Rolle's Theorem is a fundamental theorem of calculus that involves the continuity of a function and its rate of change. This theorem implies that if a function is continuous over a closed interval and differentiable over an open interval, then there will be a point in this interval on which the function’s derivative becomes 0. Let’s discuss Rolle's theorem and its geometrical interpretation.

Understanding of Rolles Theorem

Rolle’s Theorem is a rule defined for continuous function, i.e., a function that does not undergo any unexpected change or discontinuity. This theorem is named after the French mathematician Michel Rolle. It is used to find a point where the first derivative of a function becomes zero. By definition, Rolle’s Theorem states that:

“If f(x) is a continuous function on a closed interval [a, b] and f(a) = f(b), f(x) is differentiable on the open interval (a, b) then there exists at least one point c between the interval where the first derivative of f(x) is zero.” 

Rolle’s Theorem Formula

The formula for Rolle’s Theorem uses the concept of derivative because it discusses the differentiability of a function on a specific point. If a function f(x) is differentiable then by Rolle’s Theorem, it must satisfy the following conditions.

• $f(x)$ is a continuous function on the closed interval [a, b].
• The function f(x) is differentiable on the open interval (a, b).
• If f(a) is equal to f(b) then there exists a point c between the interval (a, b) where $f’(c) = 0$.

It means that the function should be continuous and differentiable to follow Rolle's Theorem.

How to apply Rolle’s Theorem?

Since it is easy to apply this theorem, there is a step-by-step way to apply this rule. These steps can assist you to find the specific point which is required for a function to be continuous. These steps are:

1. Check whether the function is continuous on the closed interval [a, b]. If the function is not continuous, you cannot proceed to the next step.
2. Calculate the values of the f(a) and f(b).
3. Find the derivative of the function as f’(x).
4. Now find the point c by equating f’(x) is zero.
5. If the f’(x) is zero on c then, Rolle's Theorem will be verified. But if the f’(x) does not equal zero then the theorem will not be verified. 

Let’s understand the application of the Rolle’s Theorem in the following example.

Rolle’s Theorem Example 1

Verify the Rolle’s Theorem for the function $y = x^2+ 1$, a = –1 and b = 1.

To verify Rolle's Theorem, the function should satisfy the three conditions. For this, we need to calculate f’(x), f(a) and f(b). The function is written as;

$y = x^2+ 1$

Since the function is a polynomial function, we can use the power rule of derivatives.

$f‘(x)=\frac{d}{dx}[x^2+1]$

$f’(x) = 2x$

Now, 

$f(-1) = f(a) = (-1)^2 +1 = 1+1$

$f(-1) = 2$

And, 

$f(1) = f(b) = 1^2+1 = 2$

Now to calculate c, equate $f’(c) = 0$

$f’(c) = 2c = 0$

Therefore, c = 0 and c belongs to (-1, 1) where f’(c) = 0. Hence Rolle's Theorem is satisfied.

Rolle’s Theorem Example 2

Verify Rolle's Theorem for the function, $f(x) = x^2-4x-3$ on the interval where a =1 and b =4.

To verify Rolle's Theorem, the function should satisfy the three conditions. For this, we need to calculate f'(x), f(a) and f(b). The function is written as;

$y = x^2+ 1$

Since the function is a polynomial function, we can use the power rule of derivatives .

$f‘(x)= \frac{d}{dx}[x^2-4x-3]$

$f'(x) = 2x – 4$

Now, 

$f(1) = f(a) = (1)^2-4(1)-3 = 1-4-3$

$f(1) = - 6$

And, 

$f(4) = f(b) = 4^2-4(4)-3 = 16-16-3 = -3$

Now to calculate c, equate f’(c) = 0

$f’(c) =2c-4 = 0$

$c = 2$

Therefore, c = 2 and c belongs to (1, 4) where f’(c) = 0. Hence Rolle's Theorem is satisfied.

Geometric Interpretation of Rolle’s Theorem

If a function f(x) satisfies all three conditions of the Rolle’s Theorem then the graphical or geometric representation can be visualized by following some simple steps. 

1. Draw a line segment that connects both points of the interval [a, b] and their corresponding functions values such as f(a) and f(b).
2. Since the end points a and b of the interval [a, b] lie on the same plane because f(a) = f(b). Then the line connecting these points will be horizontal. 
3. Draw a tangent line to the function at the point between the interval (a, b). 
4. If the tangent line intersects the horizontal line at the point c, then c will satisfy the conditions f(c) = f(a) = f(b) and f’(c) = 0.
5. If the tangent line intersects the horizontal line at more than one point. Then we have to repeat the same process with a smaller interval until we find a point where the tangent line intersects only one point. 

After using the above steps, the geometric representation will be obtained as,

Comparison between Rolle’s Theorem and Mean Value Theorem

The comparison between the Rolle’s Theorem and Mean Value Theorem can be easily analysed using the following difference table.

Conclusion

Rolle’s Theorem is a fundamental theorem in calculus that allows us to discuss the continuity of a function over a closed interval. A function satisfies Rolle’s Theorem if it satisfies the three conditions. If one of these conditions remains unsatisfied, the function does not verify Rolle’s Theorem.