Problem 46 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, graph a function on the interval \([-2,5]\) having the given characteristics. Relative minimum at \(x=-1\) Critical number at \(x=0\), but no extrema Absolute maximum at \(x=2\) Absolute minimum at \(x=5\)

Step-by-Step Solution

Answer

The function has been successfully graphed with a relative minimum at \(x=-1\), a critical number at \(x=0\) without any extrema, absolute maximum at \(x=2\), and absolute minimum at \(x=5\). The accurate graphical representation of the function requires understanding these features and using them correctly when graphing.

Step 1: Understand the Characteristics

Each of these characteristics represents a specific point on the graph of the function. A relative minimum at \(x = -1\) means the function has a lowest point in its neighborhood at \(x = -1\). A critical number at \(x = 0\) but no extrema means the function has a stationary point (the derivative equals to 0) at \(x = 0\) but it does not represent a local minimum or maximum. An absolute maximum at \(x = 2\) means the function reaches its highest point at \(x = 2\). An absolute minimum at \(x = 5\) means the function reaches its lowest point at \(x = 5\).

Step 2: Plotting the Characteristics on the Graph

First, draw your x and y-axes, labeling the x-axis with the intervals from \(-2\) to \(5\). Now, plot the points indicated by the characteristics on the graph. This would mean placing a dot at \((-1, f(-1))\) for the relative minimum, \((0, f(0))\) for the critical number, \((2, f(2))\) for the absolute maximum, and \((5, f(5))\) for the absolute minimum. Connect these points with a smooth curve while keeping in mind the behavior of functions – increasing/decreasing and concave up/down.

Step 3: Draw the Graph

Draw a smooth curve to connect the points marked in the previous step. The curve should start at the point for the relative minimum, go through the point for the critical number, reach the point of the absolute maximum, and end at the point of absolute minimum, while remaining within the interval \([-2, 5]\). The shape of the curve will be determined by the given characteristics: it should descend from the relative minimum to the critical number, ascend from there to the absolute maximum, continue descending from there to the absolute minimum, reflecting the change in the value of the function.

Key Concepts

Understanding Critical Numbers

Critical numbers are specific points on the graph of a function that are important for analyzing its shape and behavior. These points occur where the first derivative of the function is zero or undefined.
They help us understand where the graph could potentially have peaks or valleys. However, not all critical numbers lead to having local extrema.

• Calculate the first derivative of the function, and set it equal to zero to find potential critical numbers.
• Check if the derivative does not exist at some points; these points can be candidates as well.

In the given exercise, at \(x = 0\), the function has a critical number but no local extremum (neither a minimum nor a maximum). This means although the slope is zero there, the graph does not dip or peak at this point. Understanding this helps in plotting and predicting the shape of the function accurately.

Identifying Relative Minima and Maxima

Relative minima and maxima refer to points on a function graph that represent local lowest or highest values, respectively. They're not the absolute highest or lowest on the entire graph, just in their immediate vicinity.
To find these, we typically use derivatives:

• First, find the critical numbers by setting the derivative to zero or finding where it's undefined.
• Second, use the second derivative test to determine if these critical points are minima or maxima. The second derivative test checks if the graph is concave up (indicative of a minimum) or concave down (indicative of a maximum).

In our exercise, \(x = -1\) is marked as a relative minimum. This means around this point, the function dips down to a lower value compared to its neighboring points. It's crucial to graph these accurately for a clear representation of the function’s behavior.

Understanding Absolute Extrema

Absolute extrema refer to the highest or lowest points of the entire function within a given interval. They are crucial in graphing because they set the overall bound for the values the function can take.
To find absolute extrema, you should check values at critical points and endpoints of the interval.

• Evaluate the function at the critical points within the interval.
• Evaluate the function at the endpoints of the interval.
• Compare these values; the highest is the absolute maximum, and the lowest is the absolute minimum.

In our problem, the function has an absolute maximum at \(x = 2\) and an absolute minimum at \(x = 5\). This tells us that within the interval \([-2, 5]\), these are the highest and lowest values, helping us shape the graph accordingly. Understanding absolute extrema gives us insights into the full range of values the function can take in a specified interval.