Problem 36 - Calculus Early Transcendentals

Question

Let \(f\) and \(g\) be continuous on \([a, b]\) and differentiable on \((a, b) .\) Prove: If \(f(a)=g(a)\) and \(f(b)=g(b),\) then there is a point \(c\) in \((a, b)\) such that \(f^{\prime}(c)=g^{\prime}(c) .\)

Step-by-Step Solution

Answer

By Rolle's Theorem, there exists a point \(c\) in \((a, b)\) where \(f'(c) = g'(c)\).

Step 1: Understanding the Problem

We need to show that if two continuous and differentiable functions, \(f\) and \(g\), have the same values at two endpoints, \(f(a) = g(a)\) and \(f(b) = g(b)\), then there exists some point \(c\) in \((a, b)\) such that their derivatives are equal at that point.

Step 2: Setting Up the Equations

Consider the function \(h(x) = f(x) - g(x)\). Since \(f\) and \(g\) are continuous on \([a, b]\) and differentiable on \((a, b)\), \(h\) is also continuous on \([a, b]\) and differentiable on \((a, b)\).

Step 3: Applying the Endpoint Condition

From the given conditions, \(f(a) = g(a)\) and \(f(b) = g(b)\), it follows that \(h(a) = f(a) - g(a) = 0\) and \(h(b) = f(b) - g(b) = 0\).

Step 4: Using Rolle's Theorem

Since \(h(a) = h(b) = 0\), by Rolle's Theorem, there must be at least one point \(c\) in the interval \((a, b)\) such that \(h'(c) = 0\).

Step 5: Relating to Function Derivatives

Since \(h(x) = f(x) - g(x)\), we have \(h'(x) = f'(x) - g'(x)\). Thus, \(h'(c) = 0\) implies \(f'(c) - g'(c) = 0\) or equivalently \(f'(c) = g'(c)\).

Key Concepts

Continuous Functions

In the mathematical world, continuous functions are pivotal for understanding how changes in variables affect functions. A function is said to be continuous on a closed interval \[ [a, b] \] if there is no interruption in its value within that interval. Graphically, you can draw the curve without lifting your pen from the paper. This continuity implies that the function doesn't have any breaks, jumps, or holes.
Continuity is essential in calculus for evaluating limits and understanding various theorems, like Rolle’s Theorem and the Mean Value Theorem. In the context of this exercise, both functions, \( f \) and \( g \), are continuous on \[ [a, b] \]. This means:

• There are no sudden jumps between \( f(a) \) and \( f(b) \), and similarly for \( g(x) \).
• We can use tools like Rolle's Theorem because continuous functions ensure the assumptions required by these theorems hold true.

Understanding continuous functions is crucial for working with derivative concepts and other calculus principles that rely on smooth and uninterrupted behavior.

Differentiable Functions

Differentiable functions form the foundation for understanding how functions change at any point within a particular interval. A function \( f \) is differentiable on an open interval \[ (a, b) \] if it has a derivative at each point in that interval. Simply put, if a function is differentiable, you can find its rate of change or slope at every point in the interval.
Differentiability implies continuity, but remember, a continuous function is not always differentiable. Differentiable functions maintain smoother curves without sharp points or cusps within the interval of interest. In our exercise:

• Both functions \( f \) and \( g \) are differentiable on \[ (a, b) \], allowing us to calculate \( f'(x) \) for any \( x \) in this interval.
• We can confidently apply calculus tools such as derivative rules and theorems like Rolle's Theorem, which necessitate differentiability.

Differentiability is crucial for examining local behavior of functions and understanding more complex calculus concepts further in your studies.

Function Derivatives

The derivative of a function is a powerful concept in calculus that measures how a function's value changes as its input changes. For a function \( f(x) \), the derivative, denoted \( f'(x) \), provides the slope of the curve at any point \( x \). It tells us whether the function is increasing or decreasing and how quickly those changes occur.
In the context of Rolle's Theorem and our problem, we are interested in the derivatives of \( f \) and \( g \) and how they compare at specific points. Here's how we approach it:

• We construct another function, \( h(x) = f(x) - g(x) \), which represents the difference between the two functions.
• Since \( h(x) \) inherits the differentiability of \( f \) and \( g \), we can calculate \( h'(x) = f'(x) - g'(x) \).
• If at some point \( c \) in \[ (a, b) \], \( h'(c) = 0 \), then it follows that \( f'(c) = g'(c) \).

Understanding derivatives allows us to solve problems of optimization, motion, and change throughout mathematics and engineering, making it an indispensable tool in your mathematical toolkit.