Problem 22 - Calculus Early Transcendentals: Pearson New International Edition

Question

If \(f\) is continuous on \([a, b]\) and differentiable on \((a, b)\), and if \(f(a)=f(b)\), then there is at least one number \(c\) in \((a, b)\) such that \(f^{\prime}(c)=0\). Show that Rolle's Theorem is just a special case of the Mean Value Theorem. (Michel Rolle (1652-1719) was a French mathematician.)

Step-by-Step Solution

Answer

Rolle's Theorem is a special case of the Mean Value Theorem where the slope \( \frac{f(b) - f(a)}{b - a} \) is zero.

Step 1: Understand the Mean Value Theorem

The Mean Value Theorem states that if a function \( f \) is continuous on \([a, b]\) and differentiable on \((a, b)\), then there exists at least one number \( c \) in \((a, b)\) such that \( f'(c) = \frac{f(b) - f(a)}{b - a} \).

Step 2: Analyze Rolle's Theorem Assumptions

Rolle's Theorem has the additional condition that \( f(a) = f(b) \). This means that in the context of the Mean Value Theorem, the equation becomes \( f'(c) = \frac{f(b) - f(a)}{b - a} = 0 \) since \( f(a) = f(b) \) implies \( f(b) - f(a) = 0 \).

Step 3: Relate Rolle's Theorem to Mean Value Theorem

Since \( f'(c) = 0 \) is a direct consequence of the Mean Value Theorem when \( f(a) = f(b) \), we see that Rolle's Theorem can be derived as a special case where the secant slope \( \frac{f(b) - f(a)}{b - a} \) is zero.

Key Concepts

Mean Value Theorem

The Mean Value Theorem (MVT) is a fundamental concept in calculus. It provides a link between derivatives and the increase or decrease of a function over a specific interval. Let’s break it down:

• The function needs to be continuous on the closed interval \[a, b\]. This means there are no sudden jumps or gaps in the graph as you move from \(a\) to \(b\).


• The function should also be differentiable on the open interval \(a, b\). Differentiability implies that the function has a derivative at all points inside \(a\) and \(b\), which also implies smoothness.

The heart of the MVT is this: There’s at least one point \(c\) in \(a, b\) where the derivative \(f'(c)\) equals the slope of the secant line through the endpoints \((a, f(a))\) and \((b, f(b))\), illustrated by the formula:\[ f'(c) = \frac{f(b) - f(a)}{b - a}. \]
The Mean Value Theorem assures us that our function reflects some average rate of change depicted by the secant slope within the interval.

Differentiable Function

Differentiability is a property of a function that speaks about its smoothness and the ability to calculate its derivative. For a function to be differentiable on an interval \(a, b\), it must:

• Have a defined derivative at each point in the interval \(a, b\).
• Be locally linear, which means no sharp turns or cusps.

Differentiability implies continuity, but not all continuous functions are differentiable. For example, the absolute value function \( f(x) = |x| \) is continuous everywhere but not differentiable at \(x = 0\) due to the sharp corner at this point.
When talking about Rolle's Theorem and the MVT, differentiability ensures the function is smooth enough for these theorems to apply, allowing us to explore behavior like the presence of local extrema or the average slope of a function.

Continuous Function

A continuous function is one that has no interruptions, jumps, or breaks. It can be thought of as a curve that can be drawn without lifting the pencil from the paper.
For a function \(f\) to be continuous on an interval \[a, b\]:

• \(f\) must be defined at each point in \[a, b\].
• No asymptotes or open circles should exist on this interval.


• The limit approaching any point must equal the function’s value at that point.

Continuity is pivotal for applying theorems like the MVT and Rolle's Theorem, as it ensures a smooth transition over the defined interval. This smoothness is necessary to assure the existence of points where the derivative behaves in a predicted manner, such as finding points where the tangent is horizontal.

Michel Rolle

Michel Rolle was a prominent French mathematician known for Rolle's Theorem, a result now considered a particular case of the Mean Value Theorem. His contribution helped in understanding the general behavior of differentiable functions over an interval.
Rolle's Theorem posits that if a function satisfies:

• Continuity on \[a, b\]
• Differentiability on \(a, b\)
• And the condition \(f(a) = f(b)\),

then there exists at least one point \(c\) within \(a, b\) where the derivative \(f'(c) = 0\). This is visually understood as having a horizontal tangent line at some point between \(a\) and \(b\).
Rolle's Theorem is used to express scenarios where a curve must turn back at some point, provided its start and end points share the same height. This insight elegantly illustrates how calculus intertwines with geometry. Michel Rolle's work has paved the way for deeper mathematical analysis in functions and calculus.