Problem 9 - Applied Calculus

Question

Sketch the graph of a function on the inter val \(0 \leq x \leq 10\) with the given properties. Has local minimum at \(x=3\), local maximum at \(x=8\), but global maximum and global minimum at the endpoints of the interval.

Step-by-Step Solution

Answer

The graph has a global minimum at \(x=0\) and a global maximum at \(x=10\), with local minimum at \(x=3\) and local maximum at \(x=8\).

Step 1: Understand the Problem Statement

We are tasked with sketching a function graph on the interval \(0 \leq x \leq 10\). The graph should feature a local minimum at \(x=3\), a local maximum at \(x=8\), a global minimum at one endpoint (\(x=0\)) and a global maximum at the other endpoint (\(x=10\)).

Step 2: Analyze Critical Points

For a local minimum at \(x=3\), the function should be decreasing before \(x=3\) and increasing after \(x=3\). For a local maximum at \(x=8\), the function should be increasing before \(x=8\) and decreasing after \(x=8\).

Step 3: Construct the Endpoint Requirements

Since the global minimum occurs at \(x=0\), the function value at \(x=0\) must be lower than any value within the interval. Similarly, for the global maximum at \(x=10\), the function value at \(x=10\) must be greater than any value within the interval.

Step 4: Sketch the Function

Begin sketching by starting at a low point at \(x=0\) and gradually increasing towards the local minimum at \(x=3\). Then climb to the local maximum at \(x=8\), and descend slightly before rising sharply to the highest point at \(x=10\). Ensure that \(f(0) < f(3) < f(8) < f(10)\). This will satisfy the condition of the global minimum and maximum at the endpoints.

Step 5: Review the Sketch

Check that the sketch meets all the specified conditions: local minimum at \(x=3\), local maximum at \(x=8\), global minimum at \(x=0\), and global maximum at \(x=10\). The graph should visually reflect these characteristics clearly.

Key Concepts

Local Minimum

In the world of graphing functions, a local minimum is a point where the function's value is smaller than at nearby points. For example, imagine a valley in a mountain range. This valley represents a local minimum. A function has a local minimum at a certain point if as you move slightly left and right, the function values increase.
For the interval given in the problem, the local minimum occurs at \(x=3\). This means that as the function approaches \(x=3\), it decreases, reaches its lowest point here, and then starts to increase again.

• "Decreasing" before means the function value is getting smaller as \(x\) approaches \(3\).
• "Increasing" after means the function value becomes larger as \(x\) moves past \(3\).

A local minimum is not necessarily the lowest value of the function over the entire graph; it's just lower than the surrounding points.

Local Maximum

A local maximum is like the peak of a hill amid the rolling landscape of a graph. At a local maximum, the function has higher values than just to the left and right of that point. For the exercise, this peak occurs at \(x=8\).
To achieve a local maximum:

• Before \(x=8\), the function must increase, showing a rise in value as \(x\) approaches \(8\).
• After \(x=8\), the function should decrease, giving a drop in value as \(x\) moves away from \(8\).

Even though \(x=8\) is a high point compared to its neighbors, there might be even higher points on the entire graph, known as global maxima. Nevertheless, the local maximum is important for understanding the immediate behavior of the graph around \(x=8\).

Global Extrema

Global extrema refer to the absolute highest or lowest points on the entire graph, which occur at specific endpoint points in the provided interval. In the exercise, the global extrema are at the endpoints of the interval, \(x=0\) and \(x=10\).
The global minimum at \(x=0\) means that the function value at the very start of the interval is lower than any other value the function takes within the \(0 \leq x \leq 10\) range.

• The function starts at \(x=0\) with its lowest point and rises from there.

Similarly, the global maximum at \(x=10\) implies that the function value here is the highest compared to any other point in the interval.

• The function values increase after reaching local maxima until reaching the endpoint \(x=10\).

Global extrema tell us about the overall spread of the function's values throughout the interval and indicate the total range from the lowest to highest values occurring at the interval's boundaries.