Problem 16 - Essential Calculus Early Transcendentals

Question

\(15-22=\) Sketch the graph of \(f\) by hand and use your sketch to find the absolute and local maximum and minimum values of \(f .\) (Use the graphs and transformations of Sections \(1.2 . )\) $$f(x)=2-\frac{1}{3} x, \quad x \geqslant-2$$

Step-by-Step Solution

Answer

The absolute and local maximum value is \( \frac{8}{3} \) at \( x = -2 \); no minimum value exists.

Step 1: Identify the Function Type

The given function is \( f(x) = 2 - \frac{1}{3}x \), which is a linear function. Linear functions are represented by straight lines on a graph.

Step 2: Determine the Line's Attributes

The slope (\( m \)) of the line is \(-\frac{1}{3}\) and the y-intercept is \(2\). The slope tells us that the line is decreasing as \( x \) increases. The y-intercept indicates that the line crosses the y-axis at \( y = 2 \).

Step 3: Check the Domain

The function is defined for \( x \geq -2 \), which means we will only graph the line starting from the point where \( x = -2 \). For any values less than \(-2\), the function is not defined.

Step 4: Calculate Key Points for Graph

To plot the graph, calculate the y-value when \( x = -2 \). Substitute \( x = -2 \) into the function: \[ f(-2) = 2 - \frac{1}{3}(-2) = 2 + \frac{2}{3} = \frac{8}{3} \approx 2.67. \]So, one key point is \((-2, \frac{8}{3})\). Another key point can be the y-intercept, which is \((0, 2)\).

Step 5: Plot the Graph

Now, plot the key points \((-2, \frac{8}{3})\) and \((0, 2)\) on the graph. Draw a straight line through these points extending to the right, as the graph is only defined for \( x \geq -2 \). This creates a line starting at \((-2, \frac{8}{3})\) and passing through \((0, 2)\).

Step 6: Identify Maximum and Minimum

Since the line is defined starting from \( x = -2 \) and decreases continuously as \( x \) increases, the highest point on this line is at \( x = -2 \). Therefore, the absolute and local maximum value of \( f(x) \) is \( \frac{8}{3} \) at \( x = -2 \). There is no minimum value because the function decreases indefinitely as \( x \) increases.

Key Concepts

Linear Functions

Linear functions are one of the simplest forms of mathematical functions. They are typically written as a form of the equation like this: \[ f(x) = mx + b \] where \( m \) represents the slope and \( b \) is the y-intercept. The graph of a linear function is always a straight line, hence the name "linear."

• The slope \( m \): This tells us how steep the line is and in which direction it moves. A positive value means the line rises as it moves to the right, while a negative slope indicates it descends.
• The y-intercept \( b \): This is where the line crosses the y-axis. It's the value of \( f(x) \) when \( x = 0 \).

In the case of \( f(x) = 2 - \frac{1}{3}x \), the slope is \(-\frac{1}{3}\), telling us the line descends as \( x \) increases. The y-intercept is 2, indicating that the line crosses the y-axis at the point (0, 2). The line keeps descending to the right, starting from its defined domain onwards. Understanding linear functions is crucial because it is foundational to more complex mathematical concepts. It's like understanding the ABCs before reading a novel.

Absolute Maximum

In mathematics, the absolute maximum of a function is the highest point over its entire domain. It is essentially where the function reaches its peak value. For a linear function that continues infinitely, like \( f(x) = 2 - \frac{1}{3}x \), finding the absolute maximum often involves checking the endpoints of the domain. Thus, only the portion of the graph that fits within the given domain \( x \geq -2 \) is considered. This is important because it limits where the absolute maximum can be found. With the endpoint at \( x = -2 \), substituting into the function gives us \( f(-2) = \frac{8}{3} \), making it the absolute maximum value. This is because for \( x < -2 \), the function is not defined, and for \( x > -2 \), \( f(x) \) continues to decrease. An absolute maximum is crucial in real-world situations where you need to know the farthest extent something can reach, such as maximum profit or peak elevation. It directly influences decision-making processes.

Local Maximum and Minimum

Local maximum and minimum values are the peaks and troughs observed in smaller sections of a function, as opposed to the entire domain. For most linear functions, the graph is a straight line, hence it doesn't have local peaks or troughs as seen in curvy functions. However, if we focus on the restricted domain \( x \geq -2 \), \( x = -2 \) serves as a local maximum. This is the highest point within the given domain. There are no local minimum values for this function within the defined portion, as the line consistently descends without any troughs. This concept is particularly useful in analyzing segments of data or processes where the overview does not depict significant extremes, but smaller sections might have noteworthy points.

Domain and Range

The domain and range are critical factors in sketching and understanding graphs. They define the set of values that a function can take.

• The domain: It encompasses all the possible input values (\( x \)-values) for the function. For \( f(x) = 2 - \frac{1}{3}x \), the domain is restricted to \( x \geq -2 \). This means the graph of the function starts from \( x = -2 \) and extends to positive infinity.
• The range: It includes all possible output values (\( f(x) \) or \( y \)-values). For our function, as \( x \) increases from \(-2\), the \( f(x) \) values decrease indefinitely. Hence, the range is \( f(x) \leq \frac{8}{3} \) since that's the maximum value drawn at \( x = -2 \).

Understanding these concepts ensures you can accurately interpret graphs and predict how they behave. By knowing the domain and range, you can determine the limits and expectations of function behavior, helping to translate mathematical concepts into real-world understanding.