Problem 16 - Thomas Calculus

Question

In Exercises \(9-28 :\) a. Find the intervals on which the function is increasing and decreasing. b. Then identify the function's local extreme values, if any, saying where they are taken on. c. Which, if any, of the extreme values are absolute? d. Support your findings with a graphing calculator or computer grapher. $$ h(r)=(r+7)^{3} $$

Step-by-Step Solution

Answer

The function is increasing for all real numbers and has no local or absolute extrema.

Step 1: Find the derivative of the function

To find out where the function is increasing or decreasing, we first need the derivative of the function. Given the function: \[ h(r) = (r + 7)^3 \]Using the power rule, the derivative is: \[ h'(r) = 3(r + 7)^2 \]

Step 2: Determine critical points

Critical points occur where the derivative is zero or undefined. For \( h'(r) = 3(r + 7)^2 \), this is zero when:\[ 3(r + 7)^2 = 0 \]Thus solving:\[(r + 7)^2 = 0 \]\[ r + 7 = 0 \]\[ r = -7 \]So, \( r = -7 \) is a critical point.

Step 3: Test intervals around the critical points

To determine where the function is increasing or decreasing, test intervals around \( r = -7 \). Choose a test point in each interval:- Interval \((-\infty, -7):\) choose \( r = -8 \); then \( h'(-8) = 3(-1)^2 = 3 > 0 \) (increasing)- Interval \((-7, +\infty):\) choose \( r = 0 \); then \( h'(0) = 3(7)^2 = 147 > 0 \) (increasing)The function is increasing on both intervals.

Step 4: Identify local extreme values

Since the derivative stays positive on both sides of the critical point \( r = -7 \), there is no sign change, thus no local extremum. The function has no local maxima or minima.

Step 5: Determine absolute extreme values

The function \( h(r) = (r + 7)^3 \) increases without bound both as \( r \to -\infty \) and \( r \to +\infty \). Therefore, it does not have an absolute minimum or maximum value.

Step 6: Graph the function

Using a graphing calculator or software, we can confirm that \( h(r) = (r + 7)^3 \) is increasing everywhere, which agrees with our calculations. The curve continuously increases and has no local or absolute extrema. This supports our analytical findings.

Key Concepts

Derivative

When working with functions, one of the most important tools in calculus is the derivative. The derivative of a function gives us the rate at which the function's value changes with respect to changes in the variable. It's a measure of how steep the function's graph is at any point.

For the function \( h(r) = (r + 7)^3 \), we use the power rule to find its derivative. The power rule states that for any function of the form \( x^n \), the derivative is \( nx^{n-1} \). Therefore, applying this to our function, we find that:

\[ h'(r) = 3(r + 7)^2 \]

This derivative tells us that the slope of the function at any point \( r \) is determined by \( 3(r + 7)^2 \). Because it includes the square of a term, \( (r + 7)^2 \), it will always be positive or zero, indicating non-negative slope in all parts of the function.

Increasing and Decreasing Functions

Understanding whether a function is increasing or decreasing on certain intervals is key to analyzing its behavior. An increasing function rises as the input increases, while a decreasing function falls as the input grows.

For the function \( h(r) = (r + 7)^3 \), having determined the derivative to be \( h'(r) = 3(r + 7)^2 \), we can assess where the function increases or decreases by checking the sign of the derivative.

- If \( h'(r) > 0 \), the function is increasing.
- If \( h'(r) < 0 \), the function is decreasing.

For our function, since \( h'(r) = 3(r + 7)^2 \) is never negative, and equals zero only at \( r = -7 \), the function is increasing on the intervals \((- \infty, -7) \) and \((-7, +\infty)\). Thus, \( h(r) \) is overall an increasing function with no intervals of decrease.

Critical Points

Critical points of a function occur where its derivative is zero or undefined. These are the points where a function might change concavity or hit a local extremum.

For our given function \( h(r) = (r + 7)^3 \), the derivative is \( h'(r) = 3(r + 7)^2 \). Critical points occur where \( 3(r + 7)^2 = 0 \). This equation simplifies to find:

- \( (r + 7)^2 = 0 \), yielding \( r = -7 \).

Thus, the only critical point is \( r = -7 \). However, since \( h'(r) \) does not change sign around this point, there is no local extremum. It's essential when finding critical points to test the sign changes of \( h'(r) \) to confirm potential extrema.

Extreme Values

Extreme values, including local maxima and minima, reveal the highest or lowest points for a given function in certain intervals. Absolute extreme values extend this concept to the entire function domain.

In analyzing \( h(r) = (r + 7)^3 \), we observed its exclusively increasing nature due to the non-negative derivative \( h'(r) = 3(r + 7)^2 \). Therefore, this function does not have any local maxima or minima since it is never flat nor decreases at any point.

Additionally, since the function stretches toward infinity as \( r \) heads towards both \( +\infty \) and \( -\infty \), it lacks absolute extreme values, either max or min. Understanding the openness of a function's increase helps us recognize that there is no peak or valley.