Problem 22 - Thomas Calculus in SI Units

Question

Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates. $$f(x)=-x-4, \quad-4 \leq x \leq 1$$

Step-by-Step Solution

Answer

Maximum is 0 at \((-4, 0)\); Minimum is -5 at \((1, -5)\).

Step 1: Identify the Function and Interval

We are given the function \(f(x) = -x - 4\) and the interval \(-4 \leq x \leq 1\). Our goal is to find the absolute maximum and minimum values of this function over the closed interval.

Step 2: Evaluate the Function at Critical Points

The given function \(f(x) = -x - 4\) is linear, which means it does not have any critical points within the interval. This is because its derivative \(f'(x) = -1\) is constant, implying there are no points where the derivative is zero or undefined.

Step 3: Evaluate the Function at the Endpoints

Since there are no critical points, we evaluate the function at the endpoints of the interval. At \(x = -4\), \(f(-4) = -(-4) - 4 = 0\). At \(x = 1\), \(f(1) = -(1) - 4 = -5\).

Step 4: Determine the Absolute Extrema

Compare the values obtained from the endpoints. The maximum value is 0, which occurs at the point \((-4, 0)\). The minimum value is -5, occurring at the point \((1, -5)\).

Step 5: Graph the Function and Mark Extrema

Sketch the graph of \(f(x) = -x - 4\) over the interval \(-4 \leq x \leq 1\). This is a straight line with a negative slope. Mark points \((-4, 0)\) for the maximum and \((1, -5)\) for the minimum.

Key Concepts

Absolute Extrema

Absolute extrema refer to the highest and lowest points a function attains on a given interval. These points include the absolute maximum, which is the topmost point on a graph within the interval, and the absolute minimum, which is the lowest. When dealing with continuous functions on a closed interval, absolute extrema will always exist.

Finding these extrema involves several steps:

• Identify Critical Points: These are points where the derivative of the function is either zero or undefined. For linear functions like in this case, there are no critical points within the interval since the derivative is constant.
• Evaluate at Endpoints: Without critical points, you check the values of the function at the endpoints of the interval. This will help determine the absolute maximum and minimum values.

For the given function, the endpoint evaluation found the absolute maximum of 0 at (-4, 0) and the absolute minimum of -5 at (1, -5).

Linear Functions

Linear functions are equations of the form \(f(x) = mx + b\), where \(m\) and \(b\) are constants. The graph of a linear function is always a straight line.

Key features of linear functions include:

• Slope: The coefficient \(m\) represents the slope. In our example, the slope is -1, meaning the line descends as it moves from left to right.
• Y-intercept: The constant \(b\) is where the line crosses the y-axis. For the function \(f(x) = -x - 4\), the y-intercept is -4.

Since the slope is negative, this impacts how we find extrema: by merely checking the endpoints, as there are no turning points within intervals for linear functions. The absence of vertical or horizontal tangents simplifies calculations.

Graphing

Graphing functions provides a visual representation that makes understanding concepts such as slope and extrema easier. Here’s a straightforward approach to graph a linear function:

• Identify the y-intercept: Start by plotting the point where the line crosses the y-axis, which is (-4) for \(f(x) = -x - 4\).
• Use the slope: Since the slope is -1, from any point on the graph, move down 1 unit and right 1 unit to locate another point on the line.
• Draw the line: Connect these points, extending the line through the whole interval \(-4 \leq x \leq 1\).
• Mark extrema points: Highlight the highest point (-4, 0) and the lowest point (1, -5).

Graphing serves to not only locate extrema but also to better understand the nature of the function. With linear functions, the graph provides immediate insight into critical values.