Problem 15 - 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. $$ f(r)=3 r^{3}+16 r $$

Step-by-Step Solution

Answer

The function is always increasing with no local or absolute extremes.

Step 1: Find the derivative

First, differentiate the function to find \( f'(r) \). The function is \( f(r) = 3r^3 + 16r \).Differentiate term-by-term:\[ f'(r) = \frac{d}{dr}(3r^3) + \frac{d}{dr}(16r) = 9r^2 + 16. \]

Step 2: Determine critical points

Critical points occur where \( f'(r) = 0 \) or where \( f'(r) \) is undefined. Since \( f'(r) = 9r^2 + 16 \) is a polynomial, it is never undefined. Solve:\[ 9r^2 + 16 = 0 \]This equation has no real roots since 9r^2 is always non-negative and 16 is positive, so the left side never equals zero. Thus, there are no critical points.

Step 3: Analyze the derivative

Since there are no critical points and \( f'(r) = 9r^2 + 16 \) is always positive because 9r^2 is non-negative and adding 16 keeps it positive, the function is always increasing.

Step 4: Local extremes analysis

The function has no critical points, hence there are no turning points where the function changes from increasing to decreasing or vice versa. Therefore, there are no local extrema.

Step 5: Absolute extremes analysis

Given the function is always increasing, it has no minimum value as \( r \to -\infty \) and no maximum value as \( r \to \infty \). Thus, the function reaches infinity and negative infinity, and does not have absolute extreme values.

Step 6: Graphing the function

Use a graphing calculator or graphing software to plot \( f(r) = 3r^3 + 16r \). Observing the graph should confirm that the function is increasing over its entire domain without any local or absolute extreme values.

Key Concepts

Derivatives

Derivatives represent the rate of change of a function with respect to one of its variables. In calculus, the derivative of a function at a certain point describes the slope of the tangent line at that point. For the function given, \( f(r) = 3r^3 + 16r \), the derivative \( f'(r) \) helps us understand how \( f(r) \) behaves as \( r \) changes.To find the derivative, we differentiate each term separately, which for our function results in \( f'(r) = 9r^2 + 16 \). This polynomial is important because it helps determine the behavior of the initial function, such as where it increases or decreases.

Critical Points

Critical points occur where the derivative of a function is zero or undefined. These points are crucial in determining where a function might change its behavior from increasing to decreasing or vice versa.In the given exercise, we found that \( f'(r) = 9r^2 + 16 \). Since this expression is a polynomial, it is defined for all real numbers. To find critical points, we set \( f'(r) = 0 \) and solve for \( r \), which gives us the equation \( 9r^2 + 16 = 0 \).This equation does not have any real solutions because the left-hand side is always positive for any real \( r \). Consequently, the function \( f(r) \) has no critical points, meaning it doesn't switch between increasing and decreasing behavior.

Increasing and Decreasing Functions

An increasing function is one where the function value rises as the input increases, while a decreasing function does the opposite. To determine whether a function is increasing or decreasing, we look at the sign of its derivative.For the function \( f(r) = 3r^3 + 16r \), the derivative is \( f'(r) = 9r^2 + 16 \). As we found previously, this derivative is always positive because \( 9r^2 \) (being a square) is non-negative, and adding 16 prevents it from ever being zero or negative.This positive derivative tells us that \( f(r) \) is always increasing, no matter the value of \( r \). Unlike functions that alternate between increasing and decreasing based on the sign of their derivative, \( f(r) \) has a consistent upward trend across all intervals.

Local Extreme Values

Local extreme values, which include local minima and maxima, occur at critical points where a function switches from increasing to decreasing or vice versa. These points give insight into the function's peaks and troughs.In the context of our exercise, since \( f(r) = 3r^3 + 16r \) has no critical points (as discussed earlier), it cannot have any local extreme values. The absence of critical points indicates there are no peaks or troughs; the function simply keeps increasing.Without a change from increasing to decreasing at any point, \( f(r) \) lacks local minima or maxima, confirming the function's unbounded incline. This analysis, supported by a graph, shows that \( f(r) \) does not have any stopping point to examine for local extremes.