Problem 38 - Calculus Early Transcendentals: Pearson New International Edition

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 $$ \left|f\left(x_{2}\right)-f\left(x_{1}\right)\right| \leq M\left|x_{2}-x_{1}\right| $$ Note: A function satisfying the above inequality is said to satisfy a Lipschitz condition with constant \(M\). (Rudolph Lipschitz (1832-1903) was a German mathematician.)

Step-by-Step Solution

Answer

The Lipschitz condition is satisfied with constant \( M \).

Step 1: Understanding the Given Condition

We have a function \( f(x) \) defined on the interval \( (a, b) \), and its derivative satisfies \( |f'(x)| \leq M \) for all \( x \in (a, b) \). This means the rate of change of the function \( f(x) \) is bounded by \( M \).

Step 2: Expressing the Mean Value Theorem

According to the Mean Value Theorem, for a differentiable function \( f(x) \) on the interval \( (x_1, x_2) \), there exists a point \( c \in (x_1, x_2) \) such that \( f'(c) = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \).

Step 3: Applying the Inequality to the Derivative

Since \( |f'(x)| \leq M \), this inequality can be applied specifically at the point \( c \) to get \( |f'(c)| \leq M \).

Step 4: Relating the Mean Value to the Lipschitz Condition

Using the Mean Value Theorem result \( f'(c) = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \) and the bound \( |f'(c)| \leq M \), we can write:\[ \left| \frac{f(x_2) - f(x_1)}{x_2 - x_1} \right| \leq M \]

Step 5: Simplifying to Prove the Lipschitz Condition

Multiply both sides by \( |x_2 - x_1| \) to isolate the difference of functions on one side:\[ \left| f(x_2) - f(x_1) \right| \leq M |x_2 - x_1| \] which is exactly the Lipschitz condition we wanted to prove.

Key Concepts

Mean Value Theorem

The Mean Value Theorem (MVT) is a fundamental concept in calculus, particularly when discussing differentiability and continuous functions. The theorem states that if a function \( f(x) \) is continuous over a closed interval \([a, b]\) and differentiable over the open interval \((a, b)\), then there exists at least one point \( c \) in \((a, b)\) such that the derivative at that point is equal to the average rate of change of the function over \([a, b]\): \[ f'(c) = \frac{f(b) - f(a)}{b - a}. \]This provides a link between the algebraic aspect of functions (the average rate of change) and the differential aspect (the function’s derivative). The MVT implies that for any curve, there exists at least one tangent parallel to the secant line. This is crucial in proving Lipschitz conditions as it shows how the function’s derivative controls its overall behavior over intervals.

• Conditions: Continuity and differentiability are required.
• Guarantee: Existence of \( c \) where the slope of the tangent (derivative) equals the slope of the secant.

Differentiability

Differentiability of a function over an interval implies that at every point within that interval, the derivative exists. Practically, this means that the function's graph has no "sharp corners" or vertical tangents in that region. A function that is differentiable is always continuous, though the converse isn't always true. Differentiability is essential for applying the Mean Value Theorem because it ensures the existence of derivatives, a necessary condition for logical application of the theorem.
When a function is differentiable:

• The derivative at any point provides a slope of the tangent line to the function's graph at that point.
• It ensures that locally, the function behaves well and doesn't "jump."
• The presence of a bounded derivative insinuates controlled growth or change rate of the function.

Differentiability plays a key role in analyzing the behavior of functions and, in this context, establishing and proving constraints like the Lipschitz condition.

Bounded Derivative

Having a bounded derivative means the derivative \( f'(x) \) is restricted to not exceed a certain value or limit, denoted as \( M \), over the interval of interest. In simple terms, the function's rate of change does not exceed a certain speed. This is central to proving the Lipschitz condition. The Lipschitz condition essentially reflects this upper limit on how "steep" the function can be.

Consider the derivative bounded by \( M \):

• It indicates the function's slope does not surpass \( M \).
• This leads to the inequality \( \left| f'(c) \right| \leq M \), which facilitates establishing the Lipschitz condition.
• For any two points \( x_1 \) and \( x_2 \) in the interval, the difference \( \left| f(x_2) - f(x_1) \right| \) will be governed by \( M \times |x_2 - x_1| \).

Understanding bounded derivatives is pivotal to ensuring controlled behavior and understanding how a function/stretch behaves over an interval.