Problem 65 - Essential Calculus Early Transcendentals

Question

Suppose \(f\) is differentiable on an interval \(I\) and \(f^{\prime}(x)>0\) for all numbers \(x\) in \(I\) except for a single number \(c .\) Prove that \(f\) is increasing on the entire interval \(I.\)

Step-by-Step Solution

Answer

\(f\) is increasing on \(I\) due to positive derivatives everywhere except for one point.

Step 1: Understanding Differentiability and Derivatives

Since the function \(f\) is differentiable on the interval \(I\), it means that the derivative \(f'(x)\) exists for all \(x\) in \(I\). The statement \(f'(x) > 0\) for all \(x\) except one implies that the function has a positive slope everywhere except possibly at \(c\).

Step 2: Derivative Sign Interpretation

The positive derivative \(f'(x) > 0\) indicates that \(f(x)\) is increasing on any subinterval where this condition holds. Hence, \(f\) increases at all points in \(I\) except possibly at \(c\).

Step 3: Consider the Point \(c\)

At the point \(c\), \(f\) could potentially not be increasing if \(f'(c) \leq 0\). However, the problem states \(f\) is differentiable at \(c\), meaning \(f'(c)\) is defined, even though \(f'(x) > 0\) is stated as true only everywhere else. If given, analyze \(f'(c)\) directly.

Step 4: Analyze Effects of a Single Point

The fact that \(f'(x) > 0\) at all other points in the interval ensures the function is increasing elsewhere. Since a single point like \(c\) doesn't affect the overall trend of increasing across an interval, \(f\) is still considered increasing overall.

Step 5: Conclusion based on Derivative's Properties

Because \(f'(x) > 0\) except for maybe \(x = c\), \(f\) must be increasing on the entire interval \(I\). Even at a point \(c\) where \(f'(x)\) isn’t specified as positive, the presence of positive derivatives elsewhere secures its overall increasing nature.

Key Concepts

Derivative

In calculus, the derivative of a function gives us a powerful tool to understand how the function behaves at any given point.

When we have a function that is differentiable, it means that the derivative exists at every point within its interval of interest. The derivative, denoted as \( f'(x) \), tells us the rate of change or the slope of the function at any point \( x \).

Key takeaways about derivatives include:

• If \( f'(x) > 0 \), the function is increasing at that particular interval. This means as \( x \) increases, \( f(x) \) also increases.
• If \( f'(x) < 0 \), the function is decreasing, meaning \( f(x) \) decreases as \( x \) increases.
• If \( f'(x) = 0 \), the function could be constant, or it might have a local maximum or minimum.

Understanding derivatives is crucial because they provide insight into the behavior of functions beyond just their values.

Increasing Function

An increasing function is one that never falls as you move from left to right on its graph. This can be confirmed by examining the sign of its derivative. If the derivative \( f'(x) > 0 \) over an interval, the function is said to be strictly increasing on that interval.

Characteristics of an increasing function include:

• The graph of \( f \) moves upwards as \( x \) moves to the right.
• A positive derivative indicates that each subsequent value of the function is larger than the preceding one.

It’s important to note that even if \( f'(x) \) is not positive at one isolated point, the overall function can still be increasing if it has a positive derivative everywhere else. For example, having one point \( c \) where \( f'(x) \leq 0 \) doesn't change the increasing nature of the function if it's not continuous enough to affect overall trends.

Interval

Intervals in calculus are a way of describing a continuous collection of numbers. They define the domain over which we consider the derivative or behavior of a function.

Consider an interval \( I \), which is a subset of the real numbers. Function properties, such as differentiability or whether a function is increasing, are analyzed across such intervals.

• A **closed interval** is denoted by \([a, b]\) and includes the endpoints \(a\) and \(b\).
• An **open interval** is denoted by \((a, b)\) and does not include the endpoints.
• A **semi-open interval** can be either \([a, b)\) or \((a, b]\), meaning it includes only one of the endpoints.

Analyzing a function on an interval means examining all points within it to understand the function's behavior, such as where it increases or decreases, remains constant, or even where it might not be differentiable.