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Chapter 4: Problem 5

Sketch the graph of a function that has an absolute maximum, a local minimum, but no absolute minimum on [0,3].

Short Answer

Expert verified
Answer: The local minimum is at (1, 2) and the absolute maximum is at (2, 5).

Step by step solution

01

Start the graph with a local minimum

Begin by plotting a local minimum point at x=1. To do this, choose a point on the interval [0,3], let's say (1, 2). This will be our local minimum.
02

Create a continuous function that increases from the local minimum

Now, create a continuous function that increases from the local minimum point (1,2). This can be a straight line or any curve that increases as we move towards the right.
03

Plot an absolute maximum point on the interval

To plot an absolute maximum point, choose another point on the interval [0,3] with a higher y-value than our local minimum point. Let's say, the point (2, 5). This will be the absolute maximum of the function in the given interval.
04

Connect the increasing function to the absolute maximum point

The increasing function we created must now reach the absolute maximum point (2, 5). Draw a continuous curve that connects the function to this point.
05

Connect the absolute maximum point to the end of the interval

Now that we have our absolute maximum point, we need to make sure the function does not have an absolute minimum. To achieve this, create a function that handles the case x=3 but has no lower limit as x approaches the end of the interval. For example, draw a function that continuously increases again after the absolute maximum point and approaches the x=3 without settling at any limit. The graph now has an absolute maximum and a local minimum but no absolute minimum on the interval [0,3]. The final graph should have a local minimum at (1,2), an absolute maximum at (2,5), and no absolute minimum on the interval [0,3].

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