Question

Prove that if \(f: E^{1} \rightarrow E^{*}\) is of class \(\mathrm{CD}^{1}\) on \([a, b]\) and if \(-\infty\frac{f(b)-f(a)}{b-a}\left(x_{0}-a\right)+f(a) ; $$ i.e., the curve \(y=f(x)\) lies above the secant through \((a, f(a))\) and \((b, f(b)) .\)

Short answer

The function \( f(x) \) is concave, hence it lies above its secant line.

Step-by-step solution

Understand the Given Information

We are given a function \( f \) which is defined on the interval \([a, b]\) and belongs to class \( \mathrm{CD}^{1} \). This denotes that \( f \) is at least twice continuously differentiable. Additionally, we know that \( f''(x) < 0 \) for all \( x \) in \((a, b)\). This means that the function is concave on this interval.

Recall the Properties of a Concave Function

Since \( f''(x) < 0 \), this indicates that \( f' \) is strictly decreasing on \((a, b)\). A strictly decreasing derivative means that the function \( f \) itself is concave. For a concave function, any secant line will lie below the graph of the function over the interval.

Define the Secant Line

The secant line through the points \((a, f(a))\) and \((b, f(b))\) is defined by the equation: \[ y = rac{f(b) - f(a)}{b-a} (x - a) + f(a) \]. This line provides a linear interpolation between the values at endpoints \(a\) and \(b\).

Use Concavity to Compare With the Secant Line

Since \(f''(x) < 0\), \( f \) lies above this secant line. For any \( x_0 \in (a, b) \), the slope of the tangent at \( x_0 \) (\( f'(x_0) \) ) is less than the slope of the secant line. This indicates that \( f(x_0) \) is greater than the secant line at \( x = x_0 \).

Formally Prove the Inequality

We want to show that for each \( x_0 \in (a, b) \), the inequality holds: \[ f(x_0) > rac{f(b)-f(a)}{b-a}(x_0-a) + f(a) \]. Since \( f \) is concave, the secant line (average rate of change) is greater than any individual derivative \( f'(x) \), where \( x \in (a, b) \). Any tangent line will be below the function curve since tangents have steeper slopes than \( f'(x) \). By integration of these relative rates, we see the area under \( f \) from \( a \) to \( b \) being larger than under this secant line, establishing that the curve is always above the secant line.

Key concepts

Mathematical Analysis

Mathematical analysis is a broad branch of mathematics that deals with limits and related theories, such as differentiation, integration, measure, sequences, series, and analytic functions. These tools allow us to understand and describe changes, relationships, and behaviors of different functions.

It involves rigorous and precise formulation and argumentation to explore mathematical topics. For instance, in this exercise, we ascertain properties of a function and its representations like secant lines and tangents.

Mathematical analysis plays a crucial role in this context by ensuring that we can describe these properties with certainty. It helps verify that assumptions such as differentiability and concavity hold under proper mathematical scrutiny, enabling us to prove the given inequalities effectively.

• Ensures rigorous and systematic exploration of mathematical properties.
• Utilizes fundamental processes of differentiation and integration.
• Facilitates understanding of functions' representations like graphs.
• Essential for proving inequalities and relationships between functions.

Differentiability

Differentiability refers to the ability to find a derivative for a function at a given point. In simple terms, a differentiable function is one that has a tangent at any point within its domain.

In this problem, differentiability is key because we require that the function is at least twice continuously differentiable. This guarantees that the first and second derivatives exist and are continuous forms. Thus, we can use these derivatives to study the behavior of the function's curve.

When the second derivative, denoted as \( f''(x) \), is negative over an interval, it means that the function is concave. Concavity implies that as you move along the function curve, it bends downwards. This attribute is crucial for the function's relationship with its secant line, as it ensures that the function always lies above the secant.

• Ensures existence and continuity of derivatives.
• Crucial for understanding curves and their properties.
• Facilitates analysis of function behavior through derivatives.
• Key to establishing concavity of a function.

Secant Line

A secant line is a straight line that connects two points on the curve of a function. In this exercise, the secant line connects the points \((a, f(a))\) and \((b, f(b))\) on the curve of the function \(f(x)\).

The equation of this line is given by \[ y = \frac{f(b) - f(a)}{b-a} (x-a) + f(a) \]. This equation represents the average rate of change of the function between \(a\) and \(b\).

The secant line is important because it provides a benchmark to compare the function curve. If the function is concave (as in this case, due to the negative second derivative), it will always lie above the secant line. This property leads us to the conclusion that the function curve is steeper than the average rate of change indicated by the secant line for the points within the interval \((a, b)\).

• Serves as a baseline for comparing the function curve.
• Depicts average rate of change over an interval.
• Lies below the function if the function is concave.
• Essential for proving that a concave function surpasses its secant line.

Tangent Line

A tangent line is a straight line that just "touches" a function at a particular point, having the same slope as the function at that point. It's an essential tool in calculus for understanding the instantaneous rate of change of the function.

For any point \(x_0\) in \((a, b)\), the slope of the tangent line is the value of the first derivative at that point, \(f'(x_0)\). This slope represents how the function is changing at \(x_0\).

In the context of concave functions, the tangent at any point \(x_0\) on \(f(x)\) will lie below the function curve elsewhere, but it remains steeper than the slope of the secant line. This means that while the tangent touches the curve only at \(x_0\), the entire curve is higher than what a secant between two endpoints would predict.

• Represents instantaneous rate of change at a point.
• Highlights the behavior of the function curve near specific points.
• Helpful in analyzing concavity in conjunction with derivatives.
• Supports proving function surpasses its secant line by having a greater slope.