Problem 41 - Calculus Early Transcendentals: Pearson New International Edition

Question

. Prove that, if \(f\) is continuous on \(I\) and if \(f^{\prime}(x)\) exists and satisfies \(f^{\prime}(x) \geq 0\) on the interior of \(I\), then \(f\) is nondecreasing on I. Similarly, if \(f^{\prime}(x) \leq 0\), then \(f\) is nonincreasing on \(I\).

Step-by-Step Solution

Answer

If \( f' \geq 0 \), then \( f \) is nondecreasing; if \( f' \leq 0 \), then \( f \) is nonincreasing.

Step 1: Understand Definitions

To prove that a function is nondecreasing on an interval, you must show that for any two points \( x_1 \) and \( x_2 \) in the interval such that \( x_1 < x_2 \), it follows that \( f(x_1) \leq f(x_2) \). Similarly, a nonincreasing function requires \( f(x_1) \geq f(x_2) \) for \( x_1 < x_2 \).

Step 2: Mean Value Theorem Application

The Mean Value Theorem states that if \( f \) is continuous on \([a, b]\) and differentiable on \((a, b)\), then there exists a point \( c \in (a, b) \) such that \( f'(c) = \frac{f(b) - f(a)}{b - a} \). We can use this theorem if \( f' \geq 0 \) and conclude that \( f(b) - f(a) \geq 0 \), implying \( f(a) \leq f(b) \).

Step 3: Proving Nondecreasing Behavior

Assume \( x_1 < x_2 \) and apply the Mean Value Theorem on the interval \([x_1, x_2]\). Since \( f'(x) \geq 0 \) for every \( x \) in \((x_1, x_2)\), there exists a \( c \in (x_1, x_2) \) such that \( f'(c) = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \geq 0 \). Therefore, \( f(x_1) \leq f(x_2) \), hence \( f \) is nondecreasing on \( I \).

Step 4: Analogous Argument for Nonincreasing Behavior

If \( f'(x) \leq 0 \) for all \( x \) in the interior of \( I \), then by applying the Mean Value Theorem on the interval \([x_1, x_2] \) with \( x_1 < x_2 \), we have \( f'(c) = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \leq 0 \). This implies \( f(x_1) \geq f(x_2) \), and thus \( f \) is nonincreasing on \( I \).

Key Concepts

Mean Value Theorem

The Mean Value Theorem (MVT) is a fundamental concept in calculus that connects the average rate of change of a function over an interval to an instantaneous rate of change within that interval. Imagine driving a car from point A to point B. If you know your average speed was 50 km/h, the Mean Value Theorem assures you that at some moment during your trip, your speedometer showed exactly 50 km/h. This is captured by the theorem's statement:

• If a function \( f \) is continuous on a closed interval \([a, b]\) and differentiable on the open interval \((a, b)\), there exists at least one point \( c \) in \((a, b)\) such that

\[f'(c) = \frac{f(b) - f(a)}{b - a}\]This result is incredibly handy for making inferences about the behavior of a function based on its derivative. If the derivative \(f'(x)\) is always nonnegative or nonpositive in an interval, it helps us draw conclusions about the function's increasing or decreasing nature over that interval.

Continuity in Functions

Continuity is like a magical adhesive that keeps a function glued smoothly together in a given interval. What does that mean? Simply put: a function \( f \) is continuous at a point \( x = c \) if there is no abrupt jump at \( c \). In more technical terms:

• The limit as \( x \) approaches \( c \) of \( f(x) \) equals \( f(c) \) itself.
• This must hold true for every point within the interval \([a, b]\).

Why is continuity important? Continuous functions promise reliability—they don't suddenly change value without warning. This smooth behavior is crucial when applying the Mean Value Theorem, as it requires the function to be seamless over the interval in question. Imagine knitting a sweater; any knots disrupt the smooth texture much like discontinuities disrupt a function's behavior.

Nondecreasing and Nonincreasing Functions

Understanding nondecreasing and nonincreasing functions is all about how the function behaves as you move along the x-axis:

• **Nondecreasing Functions:** For any two points \( x_1 \) and \( x_2 \) within the interval, if \( x_1 < x_2 \), then \( f(x_1) \leq f(x_2) \). This means that the function doesn't decrease as you go from left to right—it either stays level or increases.
• **Nonincreasing Functions:** Similarly, if \( x_1 < x_2 \), we have \( f(x_1) \geq f(x_2) \). The function either stays the same or decreases, but it never increases.

These concepts are linked to the function's derivative. If \( f'(x) \geq 0 \) throughout an interval, the function is nondecreasing. If \( f'(x) \leq 0 \), the function is nonincreasing. This understanding helps in predicting and explaining phenomena where function values represent quantities like cost, speed, or time spent.