Question

In Exercises, find the absolute extrema of the function on the closed interval. Use a graphing utility to verify your results. $$f(x)=\frac{1}{3}(2x+5),[0,5]$$

Short answer

The absolute minimum of the given function is $\frac{5}{3}$ at $x=0$ and the absolute maximum is $5$ at $x=5$.

Step-by-step solution

Examine the Function

The function $f(x)=\frac{1}{3}(2x+5)$ is a simple linear function which is defined for all real numbers. Therefore, it has no critical points (where derivative is zero or undefined) within the interval [0,5]. Therefore, the extrema must occur at the endpoints of the interval.

Evaluate the Function at the End Points

Evaluate $f(x)$ at $x=0$ and $x=5$. For $x=0$, the function simplifies to $f(0)=\frac{1}{3}(2(0)+5)=\frac{5}{3}$. For $x=5$, the function simplifies to $f(5)=\frac{1}{3}(2(5)+5)=\frac{15}{3}=5$

Identify the Absolute Extrema

The absolute minimum of the function is the lesser of the values obtained at the end points. In this case, $f(0)=\frac{5}{3}$ is the minimum value and $f(5)=5$ is the maximum value. Hence, the absolute extrema of the function are minimum at $x=0$ and maximum at $x=5$.

Verification using a graphing utility

Plot the function $f(x)=\frac{1}{3}(2x+5)$ on a graphing utility and observe. The lowest point (minimum) and the highest point (maximum) between the interval [0,5] will confirm the analysis.

Key concepts

Linear Function

A linear function is a type of function where the graph forms a straight line. The general form of a linear function is given by $f(x)=ax+b$, where $a$ and $b$ are constants. In simpler terms, $a$ is the slope of the line, which indicates the steepness and direction. If $a$ is positive, the line slopes upwards; if negative, it slopes downwards. The $b$ is the y-intercept, the point where the line crosses the y-axis.

The function $f(x)=\frac{1}{3}(2x+5)$ is a linear function. Here, after expanding, it becomes $\frac{2}{3}x+\frac{5}{3}$, where $\frac{2}{3}$ is the slope and $\frac{5}{3}$ is the y-intercept. The slope tells us this line increases moderately as $x$ increases.

Linear functions are easy to work with due to their straightforward nature and predictable behavior. When you plot a linear function on a graph, you will always get a straight line, which can extend indefinitely in both directions. This makes finding extrema, or extreme values, straightforward when confined to a particular interval.

Endpoints

Endpoints refer to the boundary points of a given interval. In the context of this exercise, the endpoints are the specified values that define the start and stop of the interval where the function is evaluated.

For the linear function $f(x)=\frac{1}{3}(2x+5)$, and the interval $[0,5]$, the endpoints are $x=0$ and $x=5$. These endpoints are significant because they mark the limits of where we assess the function to determine its extreme values.

In exercises like this, endpoints play a crucial role since, for continuous functions on closed intervals, extrema are often found at these points. Evaluating the function at both endpoints helps identify the smallest (minimum) and largest (maximum) values within the interval range.

Closed Interval

In mathematics, a closed interval is a range of numbers that includes its endpoints. It is denoted with square brackets like $[a,b]$, indicating it contains all numbers from $a$ to $b$, including $a$ and $b$.

Closed intervals are important when finding absolute extrema because they ensure the evaluation includes the precise boundary points. In our exercise, the closed interval $[0,5]$ signifies that $x$ can take any value from 0 to 5, including both 0 and 5.

Working within a closed interval is crucial for determining extrema, as some functions may achieve their highest or lowest values precisely at these endpoints. This is especially true for linear functions, like the one in this exercise, where extrema are confirmed at the endpoints since there's no critical point within the interval.

Graphing Utility

A graphing utility is a tool or software that allows you to visually plot and analyze functions and their behaviors on a graph. It provides a graphical representation that can help verify analytical solutions, such as extrema found via calculation.

In this exercise, using a graphing utility helps confirm the findings that the minimum and maximum of the function $f(x)=\frac{1}{3}(2x+5)$ on $[0,5]$ occur at the endpoints. By plotting the function, you can see the straight line spanning from $x=0$ to $x=5$, illustrating the linear function's nature.

Modern graphing utilities allow zooming, tracing, and highlighting specific intervals and points, making it easier to confirm complex calculations. Employing graphing utilities can also help build intuition about how different functions behave and where they reach their minimum or maximum within specified intervals.