Problem 32 - 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. \(f(0)=3 ; f(2)=2 ; f(6)=0\); $$ \begin{aligned} &f^{\prime}(x)<0 \text { on }(0,2) \cup(2,6) ; f^{\prime}(2)=0 \\ &f^{\prime \prime}(x)<0 \text { on }(0,1) \cup(2,6) ; f^{\prime \prime}(x)>0 \text { on }(1,2) \end{aligned} $$

Step-by-Step Solution

Answer

The graph is decreasing on \((0, 2)\) and \((2, 6)\) with a horizontal tangent at \(x = 2\).

Step 1: Understand the Initial Values

Identify the specific points given by the initial condition: \(f(0) = 3\), \(f(2) = 2\), and \(f(6) = 0\). These will be critical points on the graph.

Step 2: Analyze the Derivative Conditions

The condition \(f'(x) < 0\) means the function is decreasing on intervals \((0, 2)\) and \((2, 6)\). The condition \(f'(2) = 0\) indicates there is a horizontal tangent at \(x = 2\).

Step 3: Analyze the Second Derivative Conditions

For \(f''(x) < 0\) on intervals \((0, 1)\) and \((2, 6)\), the graph is concave down. For \(f''(x) > 0\) on interval \((1, 2)\), the graph is concave up.

Step 4: Sketch the Graph

Begin at point \((0, 3)\). Draw a decreasing and concave down segment from \(x = 0\) to \(x = 1\). Change the concavity to concave up at \(x = 1\) until \(x = 2\), maintaining the decreasing slope, and ending at \((2, 2)\) with a horizontal tangent. Continue to \(x = 6\) with a decreasing, concave down segment, ending at \((6, 0)\).

Key Concepts

Derivative conditions

Derivative conditions help us understand how a function behaves over certain intervals. In calculus, the first derivative of a function, denoted as \(f'(x)\), gives us information about the rate of change of the function, frequently seen as its slope. When \(f'(x) < 0\), it indicates that the function is decreasing over that interval. This means the function's values are getting smaller as the input \(x\) increases.

In the original problem, the derivative condition \(f'(x) < 0\) on intervals \((0, 2)\) and \((2, 6)\) tells us that the function is decreasing in these regions. Specifically, this means as we move left to right from 0 to 2, and then from 2 to 6, the graph of the function should be moving downwards. Additionally, the condition \(f'(2) = 0\) suggests there's a horizontal tangent at \(x = 2\). This generally implies a peak or a trough, or a point of inflection, depending on second derivative conditions, which we'll explore next.

Second derivative analysis

Second derivative analysis provides insights into the concavity of the function's graph. The second derivative, \(f''(x)\), tells us how the rate of change of the slope itself is changing. In simpler terms, it helps us determine whether a curve is concave up (like a cup) or concave down (like an arch).

If \(f''(x) < 0\), the graph is concave down; this means the graph is curved like an upside-down bowl. Conversely, if \(f''(x) > 0\), the graph is concave up, indicating a cup-like curve. In the given exercise, it is stated that \(f''(x) < 0\) for intervals \((0, 1)\) and \((2, 6)\), and \(f''(x) > 0\) for \((1, 2)\). This implies the graph starts concave down from \(x = 0\) to \(x = 1\), changes to concave up from \(x = 1\) to \(x = 2\), and then returns to concave down from \(x = 2\) to \(x = 6\). These variations in concavity assist in accurately sketching the curve.

Concavity

Concavity is a crucial characteristic of a graph because it gives us important information about the shape of the graph over different intervals. Understanding concavity helps identify regions where the graph is curving upwards or downwards.

From the second derivative analysis, we know:

• The graph is concave down over \((0, 1)\) and \((2, 6)\).
• The graph is concave up over \((1, 2)\).

This specific pattern of concavity results in a visual layout where the graph begins with a downward curve, transitions into an upward curve, and then back into a downward curve. This consistent change in the graph's behavior occurs due to the specific nature of the second derivative values in the given intervals.

Function behavior

Understanding function behavior encapsulates the combined effect of derivative and concavity analyses. By considering given conditions like the values of the function at particular points, and derivative signs, we can predict and sketch how the function behaves from start to end.

In the given exercise:

• The function starts at a height of 3 at \(x = 0\), decreases to 2 at \(x = 2\), and then further decreases to 0 at \(x = 6\).
• In the interval \((0, 2)\), it is initially concave down until \(x = 1\), suggesting a typical upside-down bowl shape, and then concave up towards \(x = 2\).
• From \(x = 2\) onward till \(x = 6\), the function resumes its decreasing, concave down behavior until reaching its lowest point at \(x = 6\).

This integrated understanding allows one to comprehensively sketch the graph, honoring all critical points and conforming to the function's behavior described by derivative conditions and concavity.