Problem 74 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Let the function \(f(x)\) be continuous in the interval \([a, b]\). Prove that in this close interval there exists at least one point at which \(f(x)=\frac{f(a)+f(b)}{2}\).

Step-by-Step Solution

Answer

By defining an auxiliary function \(g(x) = f(x) - \frac{f(a) + f(b)}{2}\), we showed that this function is continuous on \([a, b]\) and changes signs at the endpoints. By applying the Intermediate Value Theorem, we concluded that there exists a point \(c\) in the closed interval \([a, b]\) such that \(g(c) = 0\), which implies that \(f(c) = \frac{f(a) + f(b)}{2}\).

Step 1: Define an auxiliary function

Let us define an auxiliary function \(g(x) = f(x) - \frac{f(a) + f(b)}{2}\). Notice that \(g(x)\) is continuous on the interval \([a, b]\), since it's the difference of two continuous functions.

Step 2: Calculate the values of the auxiliary function at the endpoints

Next, we will calculate the values of the auxiliary function at the endpoints of the interval: 1. \(g(a) = f(a) - \frac{f(a) + f(b)}{2} = \frac{f(a) - f(b)}{2}\) 2. \(g(b) = f(b) - \frac{f(a) + f(b)}{2} = \frac{-f(a) + f(b)}{2}\)

Step 3: Check the signs of the auxiliary function at the endpoints

We will now check the signs of \(g(a)\) and \(g(b)\): 1. If \(g(a)=0\), then \(f(a) = \frac{f(a) + f(b)}{2}\) and our proof is complete. 2. If \(g(b)=0\), then \(f(b) = \frac{f(a) + f(b)}{2}\) and our proof is complete. 3. If both \(g(a)\) and \(g(b)\) are non-zero, we will have: - If \(f(a) > f(b)\), then \(g(a) > 0\) and \(g(b) < 0\). - If \(f(a) < f(b)\), then \(g(a) < 0\) and \(g(b) > 0\). In either case, we have a change of signs at the endpoints of the interval for the auxiliary function \(g(x)\).

Step 4: Apply the Intermediate Value Theorem

According to the Intermediate Value Theorem (IVT), since \(g(x)\) is continuous on the interval \([a, b]\) and has a change of signs at the endpoints, there must exist at least one point \(c \in (a, b)\) such that \(g(c) = 0\).

Step 5: Conclude the proof

We now know that there exists a point \(c \in (a, b)\) such that \(g(c) = 0\). Therefore, we have: \(f(c) - \frac{f(a) + f(b)}{2} = 0 \Rightarrow f(c) = \frac{f(a) + f(b)}{2}\) We have proven that there exists at least one point \(c\) in the closed interval \([a, b]\) where \(f(c) = \frac{f(a) + f(b)}{2}\).

Key Concepts

Continuous Functions

Understanding continuous functions is pivotal in calculus as they form the foundation for the application of many theorems, including the Intermediate Value Theorem (IVT). A continuous function essentially means there are no breaks or sudden jumps in the function's graph for its entire domain.

A function is said to be continuous on an interval, like \[a, b\], if you can draw the function's graph from \(a\) to \(b\) without lifting your pencil. This implies that the function will not have any holes, gaps, or vertical asymptotes within that interval.

Why Continuity Matters

Continuity is key when applying the IVT because the theorem relies on the premise that the function has no 'jumps' in the specified interval. The concept comes into play when ensuring that for every value between the function's outputs at \(a\) and \(b\), there is a corresponding input within the interval. In simpler words, if you're traveling along the graph, every possible 'Y-value' you encounter has been or will be reached.

Auxiliary Function in Calculus

An auxiliary function in calculus is a helper tool used to facilitate the solving of problems or to provide proofs for certain properties. In many cases, an auxiliary function serves to transform a complicated problem into a simpler one, or to reveal hidden insights that make a proof more approachable.

In the context of the Intermediate Value Theorem, the auxiliary function, denoted here as \(g(x)\), is defined to subtract a constant term from \(f(x)\). This subtraction adjusts the outputs of the function so that finding a root (\(g(x)=0\)) directly relates to proving \(f(x)\) equals the average of its values at \(a\) and \(b\).

Benefits of Using an Auxiliary Function

Creating an auxiliary function is a strategic move in calculus proofs. It simplifies problems without changing the core essence. By creating a function with a specific property—like having opposite signs at two points—we can apply the IVT to find an intermediate value, such as a root, that helps us conclude our original task.

Proofs in Calculus

Proofs in calculus are logical arguments that confirm the truths of mathematical statements. These proofs often utilize definitions, theorems, and established properties to demonstrate that a particular proposition is always true. Proofs are fundamental in mathematics as they give us a reliable understanding of why formulas and theorems work the way they do.

When it comes to the Intermediate Value Theorem, the proof involves a series of logical steps, starting with the assumption that a function is continuous on the interval \[a, b\] and proceeding to show a specific value between \(f(a)\) and \(f(b)\) is achieved.

Structure of Calculus Proofs

Proofs in calculus often follow a structured approach: defining terms, stating what you aim to prove, constructing auxiliary elements (like functions), applying relevant theorems, and finally deriving the conclusion. Constructing a well-ordered proof allows others to follow the argument logically and understand the concepts at a deep level, which is the goal of any satisfying educational experience in mathematics.