Problem 26 - Thomas Calculus

Question

a. Find the open intervals on which the function is increasing and decreasing. b. Identify the function's local and absolute extreme values, if any, saying where they occur. $$ h(r)=(r+7)^{3} $$

Step-by-Step Solution

Answer

The function is increasing on \((-\infty, \infty)\). There are no local or absolute extrema.

Step 1: Find the Derivative

To determine where the function is increasing or decreasing, we first need to calculate its derivative. The function given is \( h(r) = (r+7)^3 \). Using the power rule \( \frac{d}{dr} [u^n] = n u^{n-1} \cdot \frac{du}{dr} \), where \( u = r+7 \) and \( n = 3 \), the derivative is \( h'(r) = 3(r+7)^2 \cdot 1 = 3(r+7)^2 \).

Step 2: Determine Critical Points

Critical points occur where the derivative is zero or undefined. We set the derivative to zero: \( 3(r+7)^2 = 0 \). Solving this, \( (r+7)^2 = 0 \) implies \( r+7 = 0 \), hence \( r = -7 \) is a critical point. The derivative \( 3(r+7)^2 \) is defined for all real \( r \), so there are no points where it is undefined.

Step 3: Test Intervals Around Critical Points

We determine where the function is increasing or decreasing by testing points in intervals around the critical point \( r = -7 \). Choose a test point from the interval \( (-\infty, -7) \), for example, \( r = -8 \). The derivative \( h'(-8) = 3(-8+7)^2 = 3(1) = 3 \), which is positive, meaning the function is increasing on \( (-\infty, -7) \). For \( r = -6 \) (in \( (-7, \infty) \)), \( h'(-6) = 3(-6+7)^2 = 3(1) = 3 \), also positive, which means the function is increasing on \( (-7, \infty) \).

Step 4: Identify Local and Absolute Extrema

Since the function is increasing on both \( (-\infty, -7) \) and \( (-7, \infty) \), it indicates there are no local maxima or minima, as the function doesn't change from increasing to decreasing or vice versa at \( r = -7 \). As \( r \rightarrow \infty \), \( h(r) \rightarrow \infty \), and as \( r \rightarrow -\infty \), \( h(r) \rightarrow -\infty \). Therefore, there are no absolute extrema either.

Key Concepts

Derivative

In calculus, the derivative of a function is a tool that helps us understand how the function changes at any given point. Think of it as the rate of change or the function's slope at a particular point. Calculating the derivative is essential when analyzing a function's behavior, such as determining intervals of increase or decrease. For the function given in the exercise, \( h(r) = (r+7)^3 \), we use the power rule to find the derivative. The power rule states that if you have a function in the form \( u^n \), the derivative is \( n \, u^{n-1} \, \cdot \frac{du}{dr} \).

• For \( h(r) \), let \( u = r+7 \) and \( n = 3 \).
• The derivative becomes \( h'(r) = 3(r+7)^2 \).

This derivative tells us how the function changes as \( r \) changes.

Critical Points

Critical points are values of \( r \) where the function's derivative is zero or undefined. These points are significant because they help us identify where a function might change from increasing to decreasing or vice versa, indicating potential extrema. For our function, the derivative is \( h'(r) = 3(r+7)^2 \).

• We set the derivative equal to zero: \( 3(r+7)^2 = 0 \).
• Solve for \( r \), to get \( r = -7 \).
• The derivative is defined for all real numbers, so \( r = -7 \) is our only critical point.

Identifying critical points is a crucial step in understanding a function's behavior.

Increasing and Decreasing Functions

Once we have the critical points, the next step is to determine intervals where the function is increasing or decreasing. This involves choosing test points from intervals divided by the critical point and evaluating the derivative. Here's how it works for our exercise:

• Choose \( r = -8 \) from the interval \( (-\infty, -7) \).
• The derivative \( h'(-8) = 3(1) = 3 \), which is positive, implies the function is increasing.
• Choose \( r = -6 \) from the interval \( (-7, \infty) \).
• The derivative \( h'(-6) = 3(1) = 3 \), also positive, indicates the function continues to increase in this interval.

Because \( h'(r) \) is positive on both intervals, the function is always increasing.

Local and Absolute Extrema

Local and absolute extrema refer to the highest or lowest points within a specific interval (local) or the entire domain (absolute). They are identified based on changes in the function's direction at critical points. For a function to have a local extrema, it typically changes direction.

• Our critical point, \( r = -7 \), does not exhibit a change from increasing to decreasing.
• Since \( h(r) \) is increasing on both sides of \( r = -7 \), there are no local minima or maxima.
• As \( r \to \infty \), \( h(r) \to \infty \), and as \( r \to -\infty \), \( h(r) \to -\infty \).

Therefore, \( h(r) \) has neither local nor absolute extrema, as it doesn't shift direction around any critical points or boundaries.