Problem 32 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch the graph of a continuous function fon \([0,6]\) that satisfies all the stated conditions. $$ \begin{array}{l} f(0)=3 ; f(2)=2 ; f(6)=0 \\ f^{\prime}(x)<0 \text { on }(0,2) \cup(2,6) ; f^{\prime}(2)=0 \\ f^{\prime \prime}(x)<0 \text { on }(0,1) \cup(2,6) ; f^{\prime \prime}(x)>0 \text { on }(1,2) \end{array} $$

Step-by-Step Solution

Answer

Graph starts at (0,3), decreases to (2,2) with a concave up section, then decreases to (6,0) concavely.

Step 1: Plot Given Values

Begin by plotting the points given by the conditions of the function: \(f(0) = 3\), \(f(2) = 2\), and \(f(6) = 0\). These points suggest that the graph passes through (0, 3), (2, 2), and (6, 0).

Step 2: Analyze Slope Conditions

The function's derivative \(f'(x)<0\) on intervals \((0, 2)\) and \((2, 6)\) implies that the function is decreasing in these intervals. At \(x = 2\), \(f'(2)=0\) means there is a horizontal tangent line, indicating a possible minimum or maximum.

Step 3: Analyze Concavity

The second derivative \(f''(x)<0\) on \((0, 1)\) and \((2, 6)\) suggests the graph is concave down in these regions. Conversely, \(f''(x)>0\) on \((1, 2)\) means the graph is concave up in this interval. This concavity change suggests an inflection point near \(x = 1\).

Step 4: Sketch the Graph

Using all the information: - Start at \(f(0) = 3\), - Decrease concavely to \(x = 1\), - Change to concave up until \(x = 2\) for a minimum,- Continue to decrease concavely from \(x = 2\) to \(f(6) = 0\). Ensure the slope at \(x = 2\) is zero with a smooth transition between concave up and concave down.

Key Concepts

Continuous Function

To sketch the graph of a continuous function, it is critical to understand what continuity means. A continuous function does not have any breaks, jumps, or holes in its graph between points. This characteristic means that we can draw the entire graph from start to finish without lifting the pen.

Given the points

• (0, 3)
• (2, 2)
• (6, 0)

we can say that the function passes through each of these smoothly. Ensuring continuity means confirming that the function moves naturally from one point to another without unseen interruptions.

Paying attention to continuity helps in setting up a foundation for all other properties of the graph, including how the function behaves between these specific points. Identifying intervals where the function is decreasing and recognizing transitions in behavior are easier once the function's continuity ensures smooth progressions.

Concavity and Inflection Points

Concavity tells us whether a function bends upwards or downwards. It is based on the second derivative of the function.

When

• \(f''(x)<0\) on (0, 1) and (2, 6), the graph is concave down (like a frown)
• \(f''(x)>0\) on (1, 2), the graph is concave up (like a smile)

This change in concavity often indicates an inflection point, where the graph changes from being concave down to concave up or vice versa.

As per the problem, the concavity switches at around x = 1, indicating an inflection point. Inflection points are crucial because they show where the curvature of the graph changes. Spotting them helps in accurately drawing the function's graph and understanding potential shifts in its shape.

Derivative Analysis

The first and second derivatives provide insights into the function's properties. The first derivative \(f'(x)\) tells us about the slope or rate of change of the function.

In this exercise, considering that:

• \(f'(x) < 0\) on intervals (0, 2) and (2, 6)

indicates that the function is decreasing in these intervals.

Additionally, knowing that at \(x = 2\), \(f'(2) = 0\) means the slope is zero, indicating a potential local minimum or maximum. Together with the second derivative, these insights allow us to sketch the function more precisely.

Critical Points

Critical points occur where the first derivative \(f'(x)\) is zero or undefined. These points are essential in determining the behavior of the function.

In our graph, at \(x = 2\), the derivative \(f'(2) = 0\) suggests a horizontal tangent, indicating a flat spot on the graph.

This means we could potentially observe a local minimum at this point, provided the function transitions smoothly from decreasing to increasing, informed by the concavity shifts observed earlier. These critical points help shape our understanding of where the function changes its direction or stops moving along its path for a moment.

Decreasing Intervals

A function is deemed decreasing when its graph moves downwards as we move from left to right in a certain interval.

The condition \(f'(x)<0\) on intervals (0, 2) and (2, 6) confirms this nature.

• These intervals, where the function decreases, form essential parts of the overall graph shape.

Knowing the specific regions where the function decreases allows us to predict how the function behaves more accurately. These decreasing intervals make it easier to sketch the graph by outlining segments where the function simplifies from left to right.