Problem 54 - Calculus of a Single Variable

Question

Think About It In Exercises \(53-56\) , sketch the graph of a function \(f\) having the given characteristics. $$ \begin{array}{l}{f(0)=f(2)=0} \\ {f^{\prime}(x)>0 \text { for } x<1} \\\ {f^{\prime}(1)=0} \\ {f^{\prime}(x)<0 \text { for } x>1} \\ {f^{\prime \prime}(x)<0}\end{array} $$

Step-by-Step Solution

Answer

The curve of this function starts at the point \(x=0\), arcs upwards in a concave down manner to reach a highest point at \(x=1\), and then arcs downwards in a concave down manner as well to reach the point at \(x=2\).

Step 1 - Points Where the Function Equals Zero

First plot the points where \(f(x) = 0\), which are \(x=0\) and \(x=2\).

Step 2 - Identify Where the Function is Increasing and Decreasing

Next, notice that the function \(f\) is increasing for \(x<1\) and decreasing for \(x>1\). This means that the function reaches a peak or local maximum at \(x=1\). Plot a point at \(x=1\) above the x-axis, above the two points plotted earlier.

Step 3 - Identify the Concavity of the Function

We are told that the function is concave downwards everywhere, which means that it bends downwards. This indicates that the section of the function to the left of \(x=1\) should be concaving downwards and reaching up to the point at \(x=1\), and the section of the function to the right of \(x=1\) should be starting at \(x=1\) and concaving downwards as it goes to \(x=2\).

Step 4 - Sketch the Graph

Draw a curve starting from the point at \(x=0\), concaving downwards and increasing as it reaches the point at \(x=1\), then starts decreasing also with a downward concavity as it reaches the point at \(x=2\). This curve should represent the function \(f\).

Key Concepts

Understanding Critical Points

In any function graph, a critical point is a place on the graph where the derivative, or slope, is either zero or undefined. It is crucial to identify these points, as they can indicate potential peaks or valleys on the graph. In our problem, we have a critical point at \(x = 1\) because \(f'(1) = 0\). At this point, the slope of the tangent line to the graph is flat, meaning the graph is neither increasing nor decreasing at that moment. Identifying critical points helps us understand significant changes in the behavior of the graph, guiding how we sketch it. The critical point at \(x = 1\) suggests the presence of either a local maximum or minimum, depending on the concavity and surrounding derivative signs.

Figuring Out Concavity

Concavity tells us how a curve bends on a graph. We can determine the concavity of a function by checking the sign of its second derivative \(f''(x)\). A concave down section of a graph looks like a frown (\(\cap\)), while a concave up section looks like a smile (\(\cup\)). In this problem, we are told \(f''(x) < 0\), meaning the graph is concave down everywhere. This signals that although our function might start increasing, any point that's not a point of inflection will have this concave downward shape. Thus, it's crucial for our graph to reflect this curvature at every point we draw. The consistent concavity gives us a smoother approach when sketching, aiding in accurately representing the function's behavior.

Recognizing Increasing and Decreasing Functions

The behavior of a function in terms of increasing and decreasing is examined by the first derivative \(f'(x)\). When \(f'(x) > 0\), the function is increasing, meaning it goes upward as we move along the x-axis. Conversely, when \(f'(x) < 0\), the function is decreasing, sloping downwards. In this specific situation, the function increases for \(x < 1\) and decreases for \(x > 1\). This translates to a rising curve as you approach \(x = 1\), and a descending curve as you move past this point. The change from increasing to decreasing at \(x = 1\) indicates that this could be a local extremum point. Recognizing these shifts in behavior allows one to plot a more precise graph.

Identifying a Local Maximum

A local maximum is the highest point in a neighborhood or interval, where the function changes from increasing to decreasing. It happens at a critical point if the first derivative changes sign from positive to negative. In this exercise, since \(f'(x) > 0\) for \(x < 1\) and \(f'(x) < 0\) for \(x > 1\), we have a change from positive to negative at \(x = 1\). This tells us that there is a local maximum at this point. The function reaches its peak here before starting to decline. Acknowledging where local maxima occur on a graph is important as it allows you to accurately depict important features and traits of a function. This aids significantly in sketching a graph that reflects the real behavior of our function efficiently.