Problem 53 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, sketch a graph of a function \(f\) having the given characteristics. $$ \begin{aligned} &f(0)=f(2)=0 \\ &f^{\prime}(x)>0 \text { if } x<1 \\ &f^{\prime}(1)=0 \\ &f^{\prime}(x)<0 \text { if } x>1 \\ &f^{\prime}(x)<0 \end{aligned} $$

Step-by-Step Solution

Answer

The graph should start from the point (0,0), increase to \(x=1\), peak at \(x=1\), then slope downwards continuing beyond the point (2,0).

Step 1: Plot Given Points

The first step is to mark the points (0,0) and (2,0) on a graph, as these are the only points specified by the function \(f\).

Step 2: Sketch Increasing Region

The second step is to draw a rising line from (0,0) towards \(x=1\). As \(f'(x)>0\) when \(x<1\), this shows that the function is increasing in this region.

Step 3: Sketch Decreasing Region

Next, around \(x=1\) point, the line should reach a maximum point because \(f'(1)=0\) (flat point or peak is formed when derivative is zero). Then, continue the line downwards from this peak, as \(f'(x)<0\) when \(x>1\) indicating the function is decreasing in this region.

Step 4: Complete the Graph

Lastly, ensure the downward line intersects the graph point (2,0) and continues downwards, adhering to the characteristic \(f'(x)<0\). Thus, the function's graph is an uphill curve reaching a peak at \(x=1\) and then descends, passing through the point (2,0).

Key Concepts

Function Behavior

The behavior of a function gives us clues about how the graph of the function might look. Understanding function behavior is critical when sketching graphs. We can use specific characteristics, such as where the function is increasing or decreasing, to sketch an accurate graph.
How to Determine Behavior:

• Look at the signs of the first derivative, which provide insights about where the function rises or falls.
• Identify any given points, such as intersections with the x-axis or y-axis.
• Examine the first derivative's behavior to find critical points, often where the function changes from increasing to decreasing.

In our example, the function begins at the point (0,0), travels upwards until it reaches a peak at x = 1, and then descends through the point (2,0). This function is characterized as having a maximum point, making its behavior clear thanks to the given derivative information.

Critical Points

Critical points indicate where the graph has peaks (high points) and valleys (low points). These points occur where the derivative is zero or undefined. Knowing how to find and interpret critical points is essential in graph sketching.
What are Critical Points?

• Critical points occur where the derivative, denoted as \( f'(x) \), equals zero. These are typically maxima or minima of the function.
• It can also occur where the derivative is undefined, but this isn't relevant to our example.

For the given function, it is stated that \( f'(1) = 0 \), marking a critical point at \( x = 1 \). At this point, the derivative changes from positive to negative, indicating a local maximum. This change in derivative sign shows the transition from increasing to decreasing, revealing the peak of the graph.

Derivative

The derivative provides insights into the rate of change of a function. By examining the derivative, we gain crucial information about increasing and decreasing intervals, as well as critical points.
Understanding Derivatives:

• A positive first derivative \( f'(x) > 0 \) means the function is increasing at that interval.
• A negative first derivative \( f'(x) < 0 \) indicates the function is decreasing.
• When the derivative equals zero at a point \( f'(x) = 0 \), it suggests a potential peak or valley.

In our function exercise, the derivative's signs offer a guide to sketching the graph. The interval where \( f'(x) > 0 \) tells us that the function rises to a peak. When \( f'(x) < 0 \), the function falls, creating a smooth transition confirming the graph's shape. This understanding makes sketching functions both reassuring and intuitive, grounded in the behavior encoded by the derivative.