Problem 54 - 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. $$ \begin{array}{l}{\text { Relative minimum at } x=-1} \\ {\text { Critical number (but no extremum) at } x=0} \\ {\text { Absolute maximum at } x=2} \\\ {\text { Absolute minimum at } x=5}\end{array} $$

Step-by-Step Solution

Answer

The graph starts by going downwards until reaching the relative minimum at \(x=-1\). It then increases until it reaches the critical number at \(x=0\), where it levels off without forming an extremum. After that point, it increases steeply to the absolute maximum at \(x=2\), before decreasing even more steeply to the absolute minimum at \(x=5\). The graph would look parabolic in nature respecting the given conditions.

Step 1: Understand Definitions

First, we must understand the meanings of the given characteristics. A relative minimum at a given \(x\) value means the \(y\) value is lower than the points directly around it. A critical number is a point where the first derivative is equal to zero or undefined, but in this case there is no extremum, meaning, the function does not have a maximum or a minimum at \(x=0\). An absolute maximum is the highest \(y\) value of the function, which is at \(x=2\). An absolute minimum is the lowest \(y\) value of the function on the specified interval, at \(x=5\).

Step 2: Draw the Points

Sketch the graph by drawing points for the relative minimum at \(x=-1\), critical number at \(x=0\), absolute maximum at \(x=2\), and absolute minimum at \(x=5\). These points should be at different vertical positions on the graph, ensuring they satisfy the given conditions.

Step 3: Connect the Points

Finally, join the points together with a continuous line. The line must show a decreasing trend close to \(x=-1\) representing a relative minimum, horizontal trend at \(x=0\) showing no extremum, and trends upwards and downwards moving away from \(x=2\) and \(x=5\) respectively representing an absolute maximum and an absolute minimum.

Key Concepts

Absolute Minimum

Conversely, the absolute minimum of a function on a specified interval is the lowest point to which the function descends. It's the mirror concept to the absolute maximum. Just as a mountain's peak is its absolute height, the valley floor could be considered its absolute minimum.

Our function achieves an absolute minimum at \( x=5 \), indicating that of all the values the function takes on the interval \([-2, 5]\), the value at \( x=5 \) is the lowest. Such points are vital in many fields, such as finding the point of least expense or the lowest risk in a financial portfolio.