Problem 59 - Thomas Calculus

Question

Assume that \(f\) is differentiable on \(a \leq x \leq b\) and that \(f(b)< f(a)\) Show that \(f^{\prime}\) is negative at some point between \(a\) and \(b .\)

Step-by-Step Solution

Answer

By the Mean Value Theorem, there exists a point \( c \) in \((a, b)\) where \( f'(c) < 0 \).

Step 1: Understand the Problem Statement

We are given a function \( f \), which is differentiable on the interval \([a, b]\), and is such that \( f(b) < f(a) \). Our goal is to show that the derivative \( f' \) of this function is negative at some point between \( a \) and \( b \).

Step 2: Use the Mean Value Theorem

The Mean Value Theorem states that if a function is continuous on \([a, b]\) and differentiable on \((a, b)\), then there exists some point \( c \) in \((a, b)\) such that \( f'(c) = \frac{f(b) - f(a)}{b - a} \).

Step 3: Apply the Conditions

Since \( f(b) < f(a) \), we have \( f(b) - f(a) < 0 \). Thus, \( \frac{f(b) - f(a)}{b - a} < 0 \) because \( b - a > 0 \) (since \( a < b \)).

Step 4: Conclusion from Mean Value Theorem

By the Mean Value Theorem and the calculation above, there exists some point \( c \) in \((a, b)\) such that \( f'(c) < 0 \). Thus, \( f' \) is negative at some point in the interval \((a, b)\).

Key Concepts

Differentiability

Differentiability means a function can have a derivative at every point in its domain. In simple terms, if a function is differentiable at a point, it has a well-defined tangent line at that point. For a function to be differentiable over an interval, it must be differentiable at each internal point of the interval.

Differentiability is a stronger condition than continuity. A differentiable function is always continuous, but not every continuous function is differentiable.

• An example of a function that is continuous but not differentiable is the absolute value function at zero. It has a sharp corner there, and therefore, no tangent line can be defined.
• If a function changes direction sharply at a point, it's likely not differentiable there.

Understanding differentiability is crucial for applying the Mean Value Theorem, as it requires the function to be differentiable for the theorem to hold. This ensures that we can apply the theorem to find that point where the derivative exhibits certain properties, like being negative in this exercise.

Continuous Functions

A function is continuous on an interval if you can draw its graph without lifting your pen. This means there are no breaks, jumps, or holes in the graph across that interval.

Continuity requires that small changes in the input result in small changes in the output. For a function, this means if a sequence of inputs approaches a particular point, the corresponding outputs approach the function's value at that point.

• Formally, a function \( f \) is continuous at point \( x = a \) if \( \lim_{{x \to a}} f(x) = f(a) \).
• If a function is differentiable, it automatically is continuous.

In the context of the Mean Value Theorem, continuity ensures there are no disruptions, thus making it possible to find that intermediate point where the derivative takes the value defined by the theorem.

Derivative

The derivative of a function gives us its rate of change or the slope of the tangent line at any given point. It is a fundamental concept in calculus, providing insights into the function’s behavior.

Derivatives allow us to determine several important aspects of functions:

• If the derivative at a point is positive, the function is increasing at that point.
• If the derivative is negative, the function is decreasing.
• If the derivative is zero, the function may have a local maximum or minimum, or possibly an inflection point.

In our specific problem, the derivative showing a negative value at some point between \( a \) and \( b \) indicates that the function is decreasing, which aligns with the condition \( f(b) < f(a) \). This negative derivative is crucial in proving that the function exhibits a decreasing trend over this interval, as dictated by the Mean Value Theorem.