Problem 12 - Basic Analysis: Introduction to Real Analysis

Question

Suppose \(f: I \rightarrow \mathbb{R}, g: I \rightarrow \mathbb{R},\) and \(h: I \rightarrow \mathbb{R},\) are functions. Suppose \(c \in I\) is such that \(f(c)=g(c)=h(c), g\) and h are differentiable at \(c,\) and \(g^{\prime}(c)=h^{\prime}(c) .\) Furthermore suppose \(h(x) \leq f(x) \leq g(x)\) for all \(x \in I\). Prove \(f\) is differentiable at \(c\) and \(f^{\prime}(c)=g^{\prime}(c)=h^{\prime}(c)\).

Step-by-Step Solution

Answer

The function \(f\) is differentiable at \(c\), and \(f'(c) = g'(c) = h'(c)\).

Step 1: Understand the Problem

We are given three functions, \(f\), \(g\), and \(h\), with the property that at a point \(c\), they all have the same value: \(f(c) = g(c) = h(c)\). Moreover, \(g\) and \(h\) are differentiable at \(c\), and their derivatives are equal: \(g'(c) = h'(c)\). We also know that for all \(x\) in the domain \(I\), the inequality \(h(x) \leq f(x) \leq g(x)\) holds. We are tasked to prove that \(f\) is differentiable at \(c\) and find its derivative, \(f'(c)\).

Step 2: Use the Squeeze Theorem

Using the given inequality \(h(x) \leq f(x) \leq g(x)\), along with the fact that \(f(c)=g(c)=h(c)\), we note that as \(x\) approaches \(c\), \(f(x)\) is squeezed between \(h(x)\) and \(g(x)\). According to the Squeeze Theorem, this means that the limit of the difference quotient of \(f\) will be squeezed between that of \(g\) and \(h\).

Step 3: Analyze Differentiability

Since \(g\) and \(h\) are differentiable at \(c\), their difference quotients as \(x\) approaches \(c\) equal their derivatives. Specifically, \(\lim_{x \to c} \frac{g(x) - g(c)}{x-c} = g'(c)\) and \(\lim_{x \to c} \frac{h(x) - h(c)}{x-c} = h'(c)\). Given \(g'(c) = h'(c)\), these limits are equal. Then by the Squeeze Theorem, \(\lim_{x \to c} \frac{f(x) - f(c)}{x-c} = g'(c) = h'(c).\)

Step 4: Conclude Differentiability and Equality of Derivatives

Since the limit of the difference quotient exists and equals both \(g'(c)\) and \(h'(c)\), \(f\) is differentiable at \(c\), and its derivative \(f'(c)\) is equal to both \(g'(c)\) and \(h'(c)\). Thus, \(f'(c) = g'(c) = h'(c)\) is demonstrated.

Key Concepts

Squeeze Theorem

The Squeeze Theorem is a valuable tool in calculus for finding limits. It helps us determine the limit of a function trapped, or "squeezed", between two other functions with known limits. Essentially, if two functions have the same limit at a certain point and a third function is always between them, the third function must have the same limit at that point too.
In the context of differentiability, this theorem plays a crucial role. When working with derivatives, which involve limits, the Squeeze Theorem ensures that if a function is squeezed between two differentiable functions with the same derivative at a point, then the squeezed function is also differentiable there.
In our original exercise, because \(h(x) \leq f(x) \leq g(x)\) and \(f(c) = g(c) = h(c)\), as \(x\) goes to \(c\), \(f(x)\) is trapped between \(g(x)\) and \(h(x)\). So, by the Squeeze Theorem, the derivative of \(f\) at \(c\) is the same as \(g'\)'s and \(h'\)'s at the same point.

Differentiability

Differentiability is a key component of calculus, describing how a function behaves at and around a specific point. For a function to be differentiable at a point, its derivative must exist at that point. This means the limit of the difference quotient as the input approaches the point must exist.
In our exercise, differentiability implies that \(g\) and \(h\) have well-defined slopes at \(c\), courtesy of their derivatives, \(g'(c) = h'(c)\). When trying to establish the differentiability of \(f\), using the Squeeze Theorem, we make use of the existence of \(g'(c)\) and \(h'(c)\). If \(f\) is squeezed between \(g\) and \(h\), and those derivatives equal one another, \(f\) naturally inherits this differentiability property and shares the same slope (derivative) at \(c\).
The existence of \(f'(c)\) verifies that \(f\) is smooth and changing harmoniously at point \(c\), just like \(g\) and \(h\).

Limits

Limits are the foundational blocks of calculus and are crucial for understanding concepts like continuity and differentiability. The limit of a function at a point describes the behavior of the function as it approaches that point.
In the exercise, limits are central to proving \(f\) is differentiable at \(c\). The differentiability claim for \(f\) depends on the limit of its difference quotient existing and equaling that of \(g\) and \(h\).
The difference quotient \(\frac{f(x) - f(c)}{x - c}\) should have the same limit as \(\frac{g(x) - g(c)}{x - c}\) or \(\frac{h(x) - h(c)}{x - c}\) as \(x\) approaches \(c\). This confirms \(f'(c)\) is defined by the same rate of change as \(g'(c)\) and \(h'(c)\), tying everything together via limits.

Real Analysis

Real Analysis is a branch of mathematics dealing with real numbers and real-valued sequences and functions. It provides the rigorous theoretical foundation for calculus, including concepts like limits, continuity, and differentiability.
In the context of our exercise, Real Analysis gives us the precise tools and definitions needed to prove differentiability through limits and the Squeeze Theorem. It ensures that applied methods are not just intuitive but mathematically sound.
The discipline thoroughly examines how real functions behave under various conditions, such as being squeezed, and helps us rigorously justify statements about differentiability. Real Analysis brings these intuitive calculus concepts to a level of precise argumentation and logic. With its principles, we can confidently claim that when \(f\), \(g\), and \(h\) satisfy specific conditions around \(c\), they exhibit predictable derivative behavior, as shown in the exercise solution.