Problem 33 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 29-34, sketch the graph of a continuous function fon \([0,6]\) that satisfies all the stated conditions. \(f(0)=f(4)=1 ; f(2)=2 ; f(6)=0\); $$ \begin{aligned} &f^{\prime}(x)>0 \text { on }(0,2) ; f^{\prime}(x)<0 \text { on }(2,4) \cup(4,6) ; \\ &f^{\prime}(2)=f^{\prime}(4)=0 ; f^{\prime \prime}(x)>0 \text { on }(0,1) \cup(3,4) ; \\ &f^{\prime \prime}(x)<0 \text { on }(1,3) \cup(4,6) \end{aligned} $$

Step-by-Step Solution

Answer

Sketch the function increasing from (0,1) to (2,2), then decreasing through (4,1) to (6,0), respecting given concavity.

Step 1: Identify Key Points

The function is defined on the domain \([0, 6]\). We have key points given by \(f(0)=1\), \(f(2)=2\), \(f(4)=1\), and \(f(6)=0\). These points will help us to sketch the skeleton of the graph and determine the location of local maxima or minima.

Step 2: Analyze Derivative Information

From the information given: \(f'(x) > 0\) on \((0, 2)\), indicating the function is increasing on this interval; \(f'(x) < 0\) on \((2, 4) \cup (4, 6)\), indicating the function is decreasing on these intervals. The critical points \(f'(2) = 0\) and \(f'(4) = 0\) suggest local extrema at \(x=2\) and possibly at \(x=4\).

Step 3: Analyze Second Derivative Information

\(f''(x) > 0\) on \((0, 1) \cup (3, 4)\) means the function is concave up on these intervals, which often indicates a local minimum if connected with decreasing behavior nearby. \(f''(x) < 0\) on \((1, 3) \cup (4, 6)\) indicates the function is concave down, suggesting potential points of inflection or local maxima.

Step 4: Determine Behavior of Function

Using both first and second derivative information and known points, sketch the graph. The function starts at \((0,1)\), increases towards \((2,2)\) (local maximum), decreases towards \((4,1)\) (local point with no extrma as \(f'(4) eq 0\)), then continues to decrease to \((6,0)\). Between each of these, apply the concavity information to shape the curves.

Step 5: Draw the Graph

Plot the points: \((0,1)\), \((2,2)\), \((4,1)\), and \((6,0)\). Draw a curve starting at \((0,1)\), increasing to the peak at \((2,2)\), then curving downwards through \((4,1)\), and continuing to decrease until \((6,0)\). Ensure the application of concavity matches the given derivative constraints.

Key Concepts

Continuous Functions

A continuous function is a mathematical function that has no interruptions or breaks in its graph. This means that, for every point within the function's domain, the function is defined and well-behaved without any gaps. When sketching a continuous function, it's important to ensure that the graph smoothly connects all key points given in the problem.
For instance, in our exercise, the function is defined over the interval \([0, 6]\) and specific points such as \(f(0)=1\), \(f(2)=2\), \(f(4)=1\), and \(f(6)=0\) are identified. Connecting these points smoothly without lifting your pencil while drawing the graph ensures that the function remains continuous. Therefore, when dealing with continuous functions, confirm that the graph is a single unbroken line from start to finish.

Derivatives

Derivatives measure how a function changes as its input changes. They represent the slope of the function at any given point. These values provide important insights into the behavior of the function between certain intervals. For example:

• When the first derivative \(f'(x) > 0\), the function is increasing in that region.
• When \(f'(x) < 0\), the function is decreasing.
• \(f'(x) = 0\) indicates a potential local maximum or minimum.

This information helps sketch the graph accurately. In our exercise, knowing the derivative information means recognizing where the function rises and falls, like it increases on \((0, 2)\) and decreases on \((2, 6)\) with critical points at \(x = 2\) and \(x = 4\). Understanding derivatives is crucial in determining the shape and orientation of the graph.

Graph Sketching

Graph sketching involves drawing the graph of a function based on given properties and characteristics, such as key points and derivative information. This involves:

• Plotting the key points of the function—the given values like \((0, 1)\), \((2, 2)\), \((4, 1)\), and \((6, 0)\).
• Utilizing derivative information to draw increasing and decreasing sections of the function.
• Applying concavity details to make sure the curves are drawn correctly for each segment.

This exercise guides us to draw a graph that rises to a peak at \((2, 2)\), descends through \((4, 1)\) and eventually falls to \((6, 0)\). Graph sketching combines all these elements into a coherent visual representation of the function.

Concavity

Concavity describes the direction in which a function curves. It's determined using the second derivative of the function:

• If \(f''(x) > 0\), the function is concave up, resembling a 'U' shape.
• If \(f''(x) < 0\), the function is concave down, resembling an 'n' shape.

In the context of the exercise, knowing the intervals where \(f''(x) > 0\) and \(f''(x) < 0\) helps us understand how to curve the graph appropriately between these intervals. For example, the function is concave up between \((0, 1)\) and \((3, 4)\), and concave down between \((1, 3)\) and \((4, 6)\). This distinction helps ensure that the graph's curvature reflects these mathematical properties accurately.

Local Extrema

Local extrema refer to the highest or lowest points in a specific region of a graph, often called local maximums or minimums. These points are critical because they represent significant changes in the direction of the graph. They occur where the first derivative equals zero and the second derivative provides additional confirmation:

• A local maximum occurs if \(f'(x) = 0\) and \(f''(x) < 0\).
• A local minimum occurs if \(f'(x) = 0\) and \(f''(x) > 0\).

In our exercise, the local maximum is at \((2, 2)\) since \(f'(2) = 0\) and there's a change in the slope of the function. Understanding local extrema is vital for accurately representing the function on a graph and identifying peak or valley points within a given interval.