Problem 32 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) use a graphing utility to graph the function and (b) determine the open intervals on which the function is increasing, decreasing, or constant. $$f(x)=x \sqrt{3-x}$$

Step-by-Step Solution

Answer

The function \(f(x) = x \sqrt{3 - x}\) is increasing on the open interval \(0 < x < 2\) and is decreasing on the open interval \(2 < x < 3\). The function is never constant.

Step 1: Graphing the function

Utilize a graphing utility to plot the function \(f(x) = x \sqrt{3 - x}\). You need to carefully observe the shape and position of the plot to understand the points where the function changes direction.

Step 2: Analyzing the graph

From the graphed function you can see that it starts from the point (0, 0), quickly rises to a peak indicating an increase of the function, and then decreases. This information will be valuable in the following step.

Step 3: Identifying the intervals

Given the visual representation in the graph of the function \(f(x) = x \sqrt{3 - x}\), you are able to establish intervals of increase and decrease. The function increases in the interval \(0 < x < 2\), as can be seen from the graph rising from x=0 to x=2, and then decreases in the interval \(2 < x < 3\), as can be seen from the graph falling after x=2.

Key Concepts

Understanding Increasing Intervals

When we talk about increasing intervals of a function, we are referring to the sections of the graph where the function moves upwards. This means that as the x-values get larger, the y-values also increase.
To identify these intervals, we observe the slope of the graph. If the graph is going uphill as you move from left to right, then the function is increasing in that interval.

• In our example with the function \(f(x) = x \sqrt{3-x}\), the graph rises from \(x=0\) to \(x=2\).
  This indicates an increasing interval.
• During this section, each value of \(x\) within the interval makes the function's value larger than the last.

This understanding helps predict the behavior of the function without needing to graph it every time.

Exploring Decreasing Intervals

In contrast to increasing intervals, a decreasing interval covers parts of a graph where the function's output lowers as the input increases.
This means that for each step to the right along the graph, the y-values are getting smaller.

• For the specified function, \(f(x) = x \sqrt{3-x}\), we see a drop in the graph from \(x=2\) to \(x=3\).
• This drop highlights a decreasing interval, where the function's value falls as \(x\) passes through these values.

Recognizing these intervals allows you to understand where the function is losing value and can help in various analytic scenarios, such as graph sketching or optimization problems.

Performing a Graph Analysis

Graph analysis involves looking deeper into the graph to understand more than just the rises and falls. You analyze its complete behavior such as peaks, valleys, and points of inflection.
This provides a holistic view of the function's behavior over its entire domain.

• With \(f(x) = x \sqrt{3-x}\), the graph starts at the origin \((0, 0)\), increases up to a peak, and then transitions into a decrease.
• Recognizing the turning point at \(x=2\) where the function switches from increasing to decreasing is crucial.
  This point is the maximum if we consider the local view of the graph.

By analyzing graphs, you not only understand the function's behavior at specific points but can also predict the nature of the function across a domain, just by looking at its shape and direction changes.