Problem 63 - Calculus Single Variable

Question

Suppose \(f\) and \(g\) are functions with continuous derivatives on an interval containing \([a, b] .\) Prove that if \(f(a) \leq g(a)\) and if \(f^{\prime}(x) \leq g^{\prime}(x)\) for all \(x\) in \([a, b],\) then \(f(x) \leq g(x)\) for all \(x\) in \([a, b]\)

Step-by-Step Solution

Answer

The given conditions imply \(f(x) \leq g(x)\) for all \(x\) in \([a, b]\) by analyzing \(h(x) = f(x) - g(x)\).

Step 1: Understanding the Problem

We need to prove that if two functions, \(f\) and \(g\), have continuous derivatives on an interval and satisfy \(f(a) \leq g(a)\) and \(f'(x) \leq g'(x)\), then \(f(x) \leq g(x)\) holds for all \(x\) in \([a, b]\).

Step 2: Define a New Function

Define a new function \(h(x) = f(x) - g(x)\). Given the conditions, we have \(h(a) \leq 0\).

Step 3: Derivative Analysis of h(x)

Find the derivative of \(h(x)\), which is \(h'(x) = f'(x) - g'(x)\). From the assumption, \(f'(x) \leq g'(x)\), thus \(h'(x) \leq 0\).

Step 4: Apply the Mean Value Theorem

Since \(h(x)\) is a function with a continuous derivative, use the Mean Value Theorem on \([a, x]\) for any \(x\in[a,b]\). There exists a \(c\in[a,x]\) such that \(h'(c) = \frac{h(x) - h(a)}{x - a}\).

Step 5: Conclude with Inequality

Since \(h'(x) \leq 0\), by the Mean Value Theorem, \(h(x) \leq h(a)\). Given \(h(a) \leq 0\), we find \(h(x) \leq 0\), leading to \(f(x) \leq g(x)\) for all \(x\in[a, b]\).

Key Concepts

Continuous Derivatives

In calculus, a function possessing a continuous derivative means that both the function and its derivative are well-behaved without any abrupt changes. This continuity implies two things: first that the function is smooth, and second, that there are no jumps or breaks in either the function itself or its derivative.
This concept is essential because when you apply the Mean Value Theorem or similar concepts, continuity of derivatives ensures that you can smoothly "travel" along the curve without unexpected stops. Think of it like driving on a highway without any sudden roadblocks. Everything flows precisely as it should.

• Allows usage of important theorems like the Mean Value Theorem.
• Makes sure that changes in the function are predictable and smooth.

Understanding continuous derivatives is crucial for proofs and for ensuring correct applications of mathematical theorems, such as in this exercise where the behavior of the functions over the interval relies on their continuous nature.

Inequality

In mathematics, an inequality explains how two elements compare within a mathematical sentence. Inequalities tell us which element is larger, smaller, or if they are potentially equal under particular conditions.
In this problem, we encounter inequalities like \(f(a) \leq g(a)\) and \(f'(x) \leq g'(x)\). These inequalities give us foundational relationships that must persist across the interval \([a, b]\).

• \(f(a) \leq g(a)\) implies initially, function \(f\) is not greater than \(g\).
• \(f'(x) \leq g'(x)\) tells us that the rate of change (slope) of \(f\) is not steeper than that of \(g\).

These inequalities are critical to conclude that \(f(x) \leq g(x)\). Essentially, if one function starts below another and doesn’t climb faster, it can't surpass the other.

Differentiable Functions

A function being differentiable means that it has a derivative at each point in its domain. Differentiability is not just about having a derivative, but also about having a derivative that is well-defined and finite everywhere on the interval.
In our exercise, \(f\) and \(g\) are both assumed to be differentiable over \([a, b]\). This implies:

• The functions are smooth, with no sharp corners or cusps.
• They have a derivative everywhere on the interval.

This property is significant because without differentiability, you couldn't easily compare derivatives or reliably utilize them in forming proofs. Particularly, differentiability ensures you can apply the Mean Value Theorem, as utilized in the solution, establishing necessary conditions like \(h'(x) = f'(x) - g'(x) \leq 0\), which further ensures that that \(f(x) \leq g(x)\).