Question

Prove that if \(\left|f^{\prime}(x)\right| \leq M\) for all \(x\) in \((a, b)\) and if \(x_{1}\) and \(x_{2}\) are any two points in \((a, b)\) then $$ \left|f\left(x_{2}\right)-f\left(x_{1}\right)\right| \leq M\left|x_{2}-x_{1}\right| $$

Short answer

The result follows by applying the Mean Value Theorem and using the derivative bound, \(|f'(x)| \leq M\).

Step-by-step solution

Understanding the Mean Value Theorem

The Mean Value Theorem states that if a function \( f \) is continuous on \([a, b]\) and differentiable on \((a, b)\), then there exists some \( c \in (a, b) \) such that \( f'(c) = \frac{f(b) - f(a)}{b - a} \). In this problem, \( f \) is defined on \((a, b)\). We will use this theorem to relate \( f'(c) \) with the difference \( f(x_2) - f(x_1) \).

Applying the Mean Value Theorem

Given the conditions of the problem, we can apply the Mean Value Theorem to the interval \((x_1, x_2)\). Thus, there exists some \( c \in (x_1, x_2) \) such that \( f'(c) = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \).

Using the Derivative Bound

We know from the problem statement that \( |f'(x)| \leq M \) for all \( x \) in \((a, b)\), including \( x = c \). Therefore, \( |f'(c)| \leq M \).

Relating the Function Difference to the Derivative

Using the result from Step 2 and applying the bound from Step 3, we have:\[\left|\frac{f(x_2) - f(x_1)}{x_2 - x_1}\right| = |f'(c)| \leq M.\]

Solving for the Absolute Difference

Multiply both sides by \(|x_2 - x_1|\) to solve for the absolute difference:\[ |f(x_2) - f(x_1)| \leq M|x_2 - x_1|. \] This completes the proof.

Key concepts

Calculus Proof

A calculus proof demonstrates mathematical assertions using the concepts and techniques of calculus. This involves logical reasoning and derivations based on the principles and theorems of calculus. In this particular exercise, the proof hinges on the Mean Value Theorem. The Mean Value Theorem provides a bridge between the average rate of change of a function over an interval and its instantaneous rate of change (the derivative) at some point within the interval.
Applying this theorem allows us to express the difference in function values, in terms of derivatives at some point within the interval, establishing a connection between the overall change in the function and its derivative. This serves as the backbone of our argument as we aim to prove the given inequality.

Function Differentiability

Function differentiability is a key consideration in this exercise. A function is differentiable over an interval if it has a derivative at every point within that interval. This property ensures that the function is smooth and continuous, without any breaks, sharp turns, or vertical tangent lines.
In the context of the Mean Value Theorem, differentiability on \((a, b)\) implies that the derivative \(f'(x)\) is well-defined for every \(x\) in this interval. The problem itself requires that \(f\) be differentiable over the interval to apply the theorem and find a point \(c\) where\(f'(c)\)captures the average rate of change between two function values \(f(x_1)\)and\(f(x_2)\).
Ensuring differentiability allows us to confidently apply calculus principles and derive relationships between values in the domain.

Derivative Bound

A derivative bound occurs when there is a known limit on the magnitude of the derivative of a function across an interval. This can be vital in applications where knowing how fast a function can change is crucial.
In this problem, we have \(|f'(x)| \leq M\) for all \(x\) in \((a, b)\). This restriction imposes a maximum rate of change that the function \(f\) can exhibit. When combined with the Mean Value Theorem, this bound gives us a constraint on how much the function values, \(f(x_2)\) and \(f(x_1)\), can differ from each other. Understanding this bound is essential in reaching the final inequality, showcasing the influence a known derivative limit can have on function value differences.

Absolute Difference Inequality

The absolute difference inequality refers to the comparison of the absolute values of differences to verify an established relationship, \(|f(x_2) - f(x_1)| \leq M|x_2 - x_1|\). This inequality binds the change in function values to the maximum rate of change allowed according to the derivative bound.
The steps in the solution showcase how this inequality is derived using the relationship established by the Mean Value Theorem and the known derivative bound. By showing that the function's rate of change is limited, one concludes how different two function value points can be from each other, given the maximum derivative \(M\). This conclusion effectively summarizes the essence of controlling a function's variation with respect to its input changes, reaffirming the reliance on differential calculus to establish such bounds.