Problem 45 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, graph a function on the interval \([-2,5]\) having the given characteristics. Absolute maximum at \(x=-2\) Absolute minimum at \(x=1\) Relative maximum at \(x=3\)

Step-by-Step Solution

Answer

Without having the specific values of the function at points \(x=-2\), \(x=1\), and \(x=3\), specific values cannot be assigned. However, conceptually, the graph would rise to a peak at \(x=-2\), then fall to a minimum point at \(x=1\), rise again to a local peak at \(x=3\) and then decline to ensure the maximum point is at \(x=-2\). This outline presents an idea of how the final graph would look given the characteristics outlined in the exercise.

Step 1: Identify the given points

Based on the exercise, three characteristic points are given: absolute maximum at \(x=-2\), absolute minimum at \(x=1\), and relative maximum at \(x=3\). Let's mark these points on the number line: \n\n- Absolute maximum at \(x=-2\): This is the peak of the graph, which means this point has the highest function value on the interval \([-2,5]\). \n- Absolute minimum at \(x=1\): This is the lowest point of the graph, which means this point has the lowest function value on the interval. \n- Relative maximum at \(x=3\): The graph has a local peak at this point, but this is not the highest or lowest point on the interval.

Step 2: Sketch the graph

Now, after identifying the points, a possible function that matches these characteristics can be sketched. \n\nBegin by plotting the points \((-2, f(-2))\), \((1, f(1))\), and \((3, f(3))\). By convention, let's assume the values are such that \(f(-2) > f(3) > f(1)\). Sketch the graph so that it reaches its highest point at \(x=-2\), its lowest point at \(x=1\), and has a local peak at \(x=3\). Ensure that there is a clear change in direction at all three of these points.

Step 3: Verify the result

The final step involves verification of whether the drawn graph meets the given conditions i.e., reaches its highest point at \(x=-2\), its lowest point at \(x=1\), and has a local peak at \(x=3\). Also ensure that the graph lies within the given interval \([-2,5]\).

Key Concepts

Absolute Maximum

In mathematics, an absolute maximum is the highest point over a given interval of a function. When tasked with graphing a function, determining where the absolute maximum lies within that interval is crucial. Think of it as the "peak" or "summit" of a graph.

In the exercise, the absolute maximum is specified at \(x = -2\). This means that within the interval \([-2,5]\), this point has the largest value compared to any other point on the graph.

For any function to have this absolute maximum, it must not reach a higher value elsewhere within this interval. This point can be particularly helpful in understanding the global behavior of the function as it gives one the absolute largest output based on the interval constraints.

As you practice identifying or placing absolute maxima on a graph, remember:

• Verify the interval in question.
• Ensure no other values exceed the maximum at your given point.
• Look at the endpoints of the interval to confirm they don't exceed your point of focus.

Absolute Minimum

An absolute minimum is the opposite of an absolute maximum. Instead of a peak, it identifies the lowest point over a given interval.

Within the problem’s context, the absolute minimum is found at \(x = 1\). This means that at \(x=1\), the function achieves its lowest value within the interval \([-2,5]\).

This concept becomes key when evaluating how low the function can reach, especially when assessing problems that involve optimizations or constraints.

To place or identify an absolute minimum on a graph:

• Inspect the entire interval to make sure no lower value exists.
• Consider the context of the interval’s endpoints in comparison to the point marked as minimum.
• Confirm that the curve dips or reaches a trough at the specified value, without dipping lower at any other point.

Relative Maximum

A relative maximum, also known as a local maximum, is a point where a function’s value is higher than the values of the neighboring points. Unlike absolute maximums, they aren’t necessarily the highest point on the entire interval, just a small area around it.

In this exercise, there's a relative maximum located at \(x = 3\). This means the function has a peak at this point, but there may be higher values elsewhere on the interval. It’s crucial to understand that local peaks or relative maximums often help define the shape and behavior of graphs over sections of their domains.

When dealing with relative maximums:

• Focus on the immediate neighboring points rather than the whole interval.
• Check for a change from increasing to decreasing in the graph's direction at this point.
• A function may have multiple relative maximum points, especially in larger intervals or more complex functions.