Question

In Exercises, use a graphing utility to find graphically the absolute extrema of the function on the closed interval. $$ f(x)=\frac{4}{3} x \sqrt{3-x}, \quad[0,3] $$

Short answer

Use a graphing utility to graph the function \(f(x)=\frac{4}{3} x \sqrt{3-x}\) on the interval \([0,3]\). The highest and lowest points on the graph represent the absolute maximum and minimum respectively, and the coordinates of those points indicate the position and the value of these extrema.

Step-by-step solution

Understanding the Function

The given function \(f(x)=\frac{4}{3} x \sqrt{3-x}\) is a simple function defined in \([0,3]\), meaning it only has real values in this interval. You calculate function values for inputs within this interval.

Graph the Function

Using any graphing utility (like a calculator or a computer software), graph the function \(f(x)=\frac{4}{3} x \sqrt{3-x}\) on the interval \([0,3]\). This will enable a clear visualization of where the function reaches its highest and lowest points.

Find the Absolute Extrema

The highest point on the graph in the closed interval \([0,3]\) is the absolute maximum of the function \(f(x)\), while the lowest point is the absolute minimum. The graphing utility should provide a tool to find these points exactly.

Interpret the Results

The exact coordinates of these points (x,y) are the absolute extrema. The x-coordinate is the input where the function reaches its extreme value, and the y-coordinate is the value of the function at that point.

Key concepts

Graphing Utility

A graphing utility is an essential tool used to visualize mathematical functions. This can refer to a variety of softwares or electronic tools, such as graphing calculators, online graphing websites, or programs like Desmos or GeoGebra. These utilities help in plotting the function for a specified interval, allowing you to see the behavior of the function at various points.

When you input the function and the interval into a graphing utility, it creates a visual graph. This visual representation is crucial, as it allows you to quickly locate the maxima and minima, or the highest and lowest points, on your graph. By using a graphing utility, the tedious process of calculating multiple points manually is avoided, and instead you gain immediate insight into the overall shape and key features of the function in question.

• Visualize complex functions easily.
• Identify key points like maxima or minima.
• Save time by avoiding manual calculations.

Absolute Extrema

The absolute extrema of a function are key points where the function reaches its absolute highest or lowest value. In the context of a graph, these are often referred to as the absolute maximum or minimum. Determining these points is crucial for understanding the full range of a function's output.

To find the absolute extrema, especially when using a graphing utility, look for the peaks and troughs within the given interval. The highest peak represents the absolute maximum, while the lowest trough represents the absolute minimum. With a graphing utility, these points can be pinpointed accurately without relying entirely on algebraic methods.

• Absolute maximum is the highest point on the graph.
• Absolute minimum is the lowest point.
• Critical for understanding function behavior over specified intervals.

Closed Interval

A closed interval is a range of x-values where the function is specifically evaluated, including the endpoints. Represented as \([a, b]\), a closed interval implies that every value between \('a'\) and \('b'\), including both endpoints, is considered in the analysis of the function.

In our exercise, the interval \([0, 3]\) instructs us to focus only on the input values between 0 and 3. This means that the behavior of the function outside this range is not analyzed, which simplifies the task by narrowing down to the essential parts of the function relevant to finding the extrema. The closed interval ensures that the points at the edges are not ignored, which can sometimes represent significant extrema themselves.

• Involves all points between two endpoints including the endpoints themselves.
• Helps focus analysis on a specific range of interest.
• Often useful in finding meaningful extrema within these bounds.