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

The function \(f(x)\) is increasing on the interval \((a, b)\).

Step 1: Understanding the Problem

We need to prove that the function \(f(x)\) is increasing over the interval \((a, b)\) given that its derivative \(f'(x)\) is greater than zero everywhere except at one point \(x_0\) in \((a, b)\). This implies that \(f'(x) > 0\) on the intervals \((a, x_0)\) and \((x_0, b)\).

Step 2: Consider the Interval \((a, x_0)\)

On the interval \((a, x_0)\), the derivative \(f'(x)\) is positive. By the definition of a derivative, if \(f'(x) > 0\) on an interval, then \(f(x)\) is increasing on that interval. Therefore, \(f\) is increasing on \((a, x_0)\).

Step 3: Consider the Interval \((x_0, b)\)

On the interval \((x_0, b)\), the derivative \(f'(x)\) is also positive. Similarly, since \(f'(x) > 0\) on this interval, \(f(x)\) is increasing on \((x_0, b)\).

Step 4: Consider Point \(x_0\) Separately

At the point \(x_0\), the condition given does not imply that \(f'(x)\) must be positive or even defined. However, since \(f\) is continuous and increasing on either side of \(x_0\), \(f\) does not decrease at \(x_0\) and \(f\) remains continuous.

Step 5: Conclusion about the Entire Interval \((a, b)\)

Since \(f(x)\) is increasing on \((a, x_0)\) and \((x_0, b)\), and continuous at \(x_0\) with no decrease, \(f(x)\) is increasing on the entire interval \((a, b)\).

Key Concepts

Continuity

Continuity is a fundamental concept in calculus that indicates how functions behave between two points. A function \(f\) is continuous on an interval \((a, b)\) if it has no breaks, holes, or jumps within this interval. More technically, for every point \(c\) within the interval, the limit of \(f(x)\) as \(x\) approaches \(c\) from both the left and right must equal \(f(c)\).
When a function is continuous in this way, it ensures that you can draw its graph without lifting your pen off the paper. Continuous functions are quite predictable because they don't have unexpected behavior within the interval.

• Continuity at a point means \(\lim_{x\to c} f(x) = f(c)\)
• If \(f\) is continuous on \((a, b)\), it behaves smoothly over this entire interval
• Continuity ensures that functions are well-behaved and enables us to apply various theorems like the Intermediate Value Theorem

In the given problem, the continuity of the function \(f\) on \((a, b)\) is crucial as it underpins the behavior of the function at the single exception point \(x_0\).

Derivative

The derivative is a cornerstone of calculus. It measures how a function changes as its input changes, essentially giving us the slope of the tangent line to the function at any point. When you see \(f'(x) > 0\), it means that the function \(f(x)\) is increasing at that point since the slope of the tangent is positive.
Derivatives can tell us a lot about the behavior of functions:

• Indicates the rate of change of the function at a given point
• A positive derivative signifies an increasing function, while a negative one shows a decreasing function
• Zero derivative at a point suggests a potential local maximum or minimum

In our problem, the positive derivative across intervals \((a, x_0)\) and \((x_0, b)\) along with the continuity is what helps us conclude that \(f(x)\) is increasing on \((a, b)\). Even though \(f'(x)\) may not be defined exactly at \(x_0\), the assumption is acceptable due to continuity around that point.

Increasing Function

An increasing function is one where, as the input values move from left to right, the output values rise. Mathematically, a function \(f(x)\) is increasing on an interval if for every pair of numbers \(x_1 < x_2\) in the domain, \(f(x_1) < f(x_2)\).
Why do we care about increasing functions?

• They help in understanding function behavior over intervals
• Increasing functions ensure that as you move through the interval, the function never decreases
• Such functions lead to predictable outputs, useful in modeling and calculations

In the given exercise, the positive nature of the derivative indicates the increasing nature of \(f(x)\) on the intervals \((a, x_0)\) and \((x_0, b)\). The continuity ensures that there is no interruption in the increasing pattern at point \(x_0\). Both the derivative behavior and continuity help assert that \(f\) remains increasing over \((a, b)\).

Interval Analysis

Interval analysis in calculus involves breaking down the domain of a function into manageable sections for easier evaluation. By considering smaller intervals, one can study the function's behavior in segments rather than its entirety at once.
Here’s why interval analysis is indispensable:

• Allows us to apply different rules and theorems on smaller portions
• Simplifies complex problems into easier tasks
• Offers clarity when a function behaves differently over segments

In the problem at hand, analyzing \(f(x)\) separately on \((a, x_0)\) and \((x_0, b)\) helps us leverage the fact that \(f'(x) > 0\) in these individual segments. This ensures that \(f\) is increasing on both and remains continuous, making it easier to conclude about the entire interval \((a, b)\). By using interval analysis, we ensure the entire piecewise behavior of \(f(x)\) is effectively communicated.