Problem 53 - Calculus of a Single Variable

Question

Creating the Graph of a Function In Exercises 53 and \(54,\) graph a function on the interval \([-2,5]\) having the given characteristics. $$ \begin{array}{l}{\text { Absolute maximum at } x=-2} \\ {\text { Absolute minimum at } x=1} \\ {\text { Relative maximum at } x=3}\end{array} $$

Step-by-Step Solution

Answer

The graph should show a high peak at \(x=-2\) (absolute maximum), a low minimum point at \(x=1\) (absolute minimum), and another peak at \(x=3\) (relative maximum). The function value should decrease after the point \(x=3\).

Step 1: Interpret the characteristics

The first step is understanding what each of these characteristics mean. An absolute maximum at \(x=-2\) means that the function reaches its highest point at \(x=-2\). The absolute minimum at \(x=1\) means that the function has its lowest value at \(x=1\). A relative maximum at \(x=3\) suggests that the function reaches a maximum point at \(x=3\), but not the highest point on the entire function.

Step 2: Draw general shape of the graph

Start by drawing a curve that starts from a peak at \(x=-2\), goes down to a minimum at \(x=1\) and goes up again to a relative maximum at \(x=3\). Then decrease the value of the function after \(x=3\). The absolute maximum should be the highest point and the absolute minimum should be the lowest point on the graph.

Step 3: Refine the graph

The final step is to refine the graph by making sure all the curves are smooth and the maximums and minimums are clearly at the specified x values. If you can, try to extend the graph on either side of the x-axis to show that the maximum at \(x=-2\) and minimum at \(x=1\) are indeed absolute values.

Key Concepts

Understanding Absolute Maximum

In a function, the absolute maximum is the highest point over the entire domain of the function. This value is crucial in determining the peak of the graph and gives a clear idea of the range within which all other values of the function fall. For a function graph, the absolute maximum can be visualized as the tallest point on the curve when graphically represented.
The term "absolute" implies that no other point within the defined interval or domain is higher than this maximum. Specifically, in our exercise, the absolute maximum occurs at \(x = -2\). This means that on the graph, the point where \(x = -2\) will be the topmost point.

• The curve reaches this point and no other peaks exist beyond this level within the interval.
• This point is important for determining the overall spread and behavior of the function within the analyzed range.

When sketching a function, ensuring that this absolute maximum is accurately plotted will help in shaping the rest of the graph, confirming that it adheres to this characteristic closely.

Exploring Absolute Minimum

The absolute minimum of a function is the lowest point that the function reaches over its entire domain. This key concept indicates where the function graph hits its lowest valley. It's the point where, within the specified interval, the function attains its least value, offering perspective on the extent of variation the function graph can experience.
In our particular case, the function has an absolute minimum at \(x = 1\). This tells us that on the graph, \(x = 1\) corresponds to the lowest dip the curve will take.

• The curve will never fall below this point while within the defined interval.
• It's a vital aspect when considering the range and the general trend the function follows.

During the drawing process, properly placing the absolute minimum helps in correctly forming the wave and ensuring all subsequent fluctuations stay true to the minimum condition set by the exercise.

Clarifying Relative Maximum

A relative maximum, sometimes called a local maximum, is a point where the function reaches a peak within a certain neighborhood but not necessarily the overall highest point in the function's domain. This subtly different maximum indicates localized behavior and is critical when analyzing fluctuations or smaller segments of a function's graph.
For our exercise, \(x = 3\) is noted as a relative maximum. This means that around \(x = 3\), the function is higher than the surrounding values but not as high as the absolute maximum.

• This peak does not contradict other maximum criteria like the absolute maximum.
• It's particularly useful in understanding where minor peaks occur and how they fit into the curve's full picture.

While graphing, a relative maximum shows a localized rise in the curve, and it helps in portraying realistic undulations in the graph, balancing between the absolute extrema points noted.