Problem 25 - Calculus Early Transcendentals: Pearson New International Edition

Question

Prove: If \(f\) is continuous on \((a, b)\) and if \(f^{\prime}(x)\) exists and satisfies \(f^{\prime}(x)>0\) except at one point \(x_{0}\) in \((a, b)\), then \(f\) is increasing on \((a, b) .\) Hint: Consider \(f\) on each of the intervals \(\left(a, x_{0}\right]\) and \(\left[x_{0}, b\right)\) separately.

Step-by-Step Solution

Answer

Function \( f \) is increasing on \((a, b)\).

Step 1: Define the Problem and Given Conditions

We have a function \( f \) that is continuous on the interval \((a, b)\) and differentiable everywhere except possibly at a single point \( x_0 \). We know \( f' (x) > 0 \) for all \( x eq x_0 \) in \( (a, b) \). Our task is to show that \( f \) is increasing on \( (a, b) \).

Step 2: Analyze Interval \( (a, x_0] \)

On the interval \((a, x_0] \), \( f \) is continuous, and \( f' (x) > 0 \) whenever \( x eq x_0 \). By the mean value theorem, for any points \( x_1, x_2 \in (a, x_0] \) with \( x_1 < x_2 \), there exists \( c \in (x_1, x_2) \) such that \( f(x_2) - f(x_1) = f'(c)(x_2 - x_1) \). Since \( f'(c) > 0 \), it follows that \( f(x_2) > f(x_1) \), indicating that \( f \) is increasing on \( (a, x_0] \).

Step 3: Analyze Interval \( [x_0, b) \)

Similarly, on the interval \([x_0, b)\), since \( f \) is differentiable at all points except potentially at \( x_0 \) and \( f'(x) > 0 \) for every \( x eq x_0 \), the mean value theorem applies to any points \( x_1, x_2 \in [x_0, b) \) with \( x_1 < x_2 \). This implies \( f(x_2) - f(x_1) = f'(c)(x_2 - x_1) \) for some \( c \in (x_1, x_2) \), and thus \( f(x_2) > f(x_1) \). Therefore, \( f \) is increasing on \([x_0, b)\).

Step 4: Conclude for Entire Interval \((a, b)\)

Since \( f \) is shown to be increasing on both \( (a, x_0] \) and \( [x_0, b) \), and since these two intervals together cover \((a, b)\), we conclude that \( f \) is increasing on the entire interval \( (a, b) \). The point \( x_0 \) does not disrupt the increasing nature of \( f \) because the derivative \( f' \) is positive around it and continuity ensures no sudden changes.

Key Concepts

Mean Value Theorem

The Mean Value Theorem (MVT) is a fundamental concept in calculus that connects the value of a function's derivative to the function's average rate of change over an interval. It states that if a function \( f \) is continuous on the closed interval \([a, b]\) and differentiable on the open interval \((a, b)\), then there exists at least one point \( c \) in \((a, b)\) such that:
\[ f'(c) = \frac{f(b) - f(a)}{b - a}. \]
This theorem is particularly useful in understanding the behavior of functions, as it guarantees the existence of a tangent with the same slope as the secant line that joins the endpoints \( (a, f(a)) \) and \( (b, f(b)) \). When applied, the MVT helps to demonstrate periods of increase or decrease within the function based on its derivative.

• Continuity on \([a, b]\) ensures there are no jumps or breaks.
• Differentiability on \((a, b)\) means the function has a defined tangent at every point.

Differentiability

Differentiability of functions is a key concept that describes the existence of a function's derivative at a point. If a function \( f \) is differentiable at a point \( x_0 \), it means that:
\( f'(x_0) = \lim_{h \to 0} \frac{f(x_0+h) - f(x_0)}{h} \)
exists. Differentiability implies continuity—it ensures the function behaves predictably as it changes. However, it's crucial to note that while continuity is necessary for differentiability, it is not sufficient—functions that are continuous might not always be differentiable.

• A curve having no sharp corners or cusps is typically differentiable.
• If a function isn't smooth or has abrupt changes in direction, it may lack differentiability.

For a function to be differentiable on an interval \((a, b)\) except potentially at one point, it suggests that the lone point might pose a challenge to the existence of a tangent, yet won't necessarily affect the overall trend of the function, such as being increasing.

Increasing Functions

An increasing function is one where, as we move from left to right along the x-axis, the function values (y-values) rise. Formally, a function \( f \) is called increasing on an interval \( (a, b) \) if, for any two points \( x_1, x_2 \) within that interval, whenever \( x_1 < x_2 \), then \( f(x_1) \leq f(x_2) \).
For differentiable functions, the sign of the derivative provides a simple test for increasing behavior.

• If \( f'(x) > 0 \) for every \( x \) in \((a, b)\), \( f \) is strictly increasing on that interval.
• If \( f'(x) \geq 0 \), it is non-decreasing.

In cases like the given exercise, recognizing \( f'(x) > 0 \) across an interval (except possibly at a single point) is instrumental in proving the function is increasing, as it shows the slope of tangents remains positive. It confirms a continuous upward trend unaffected by that isolated point where the derivative's sign is uncertain.

Continuity and Differentiation

Continuity and differentiation are two intertwined concepts that play a critical role in analyzing functions. A function is continuous over an interval if there's no interruption, disjoint, or gap in its graph within that range. This continuous nature is foundational because, without it, differentiation fails.When assessing a problem like whether a function is increasing, continuity ensures the absence of jumps in the function's behavior. Differentiation, on the other hand, provides insights into how that behavior changes.

• Continuity guarantees the function can be drawn without lifting the pen.
• Differentiation allows us to measure how quickly or slowly the function climbs or dips.

In the specific scenario where a function is continuous over \( (a, b) \) and differentiable except at a point \( x_0 \), the smooth nature of \( f \) ensures that small changes in \( x \) don't lead to radical changes in \( f(x) \). Combined, these properties imply a predictable evolution of the function across the interval, pivotal in proving characteristics like increasing behavior despite exceptions.