Problem 30 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch the graph of a function with the given properties. \(f\) is differentiable, has domain \([0,6]\), reaches a maximum of 4 (attained when \(x=6\) ) and a minimum of \(-2\) (attained when \(x=1\) ). Additionally, \(x=2,3,4,5\) are stationary points.

Step-by-Step Solution

Answer

The function's graph starts at a value greater than -2, decreases to -2 at \(x=1\), has local changes between \(x=2\) and \(x=5\), and finally increases to 4 at \(x=6\).

Step 1: Identify Key Points

The function has a maximum value of 4 at \(x=6\) and a minimum value of \(-2\) at \(x=1\). Additionally, it has stationary points at \(x=2, 3, 4,\) and \(5\). These points and values are essential ingredients for sketching the graph.

Step 2: Consider Stationary Points

The stationary points \(x=2, 3, 4,\) and \(5\) suggest that the derivative of the function \(f'(x) = 0\) at these points. This means that at each of these \(x\)-coordinates, the function will have a horizontal tangent line.

Step 3: Analyze Function Behavior

From \(x=0\) to \(x=1\), the function must decrease to reach its minimum value due to the decreasing slope. At \(x=1\), it touches -2. Between \(x=1\) and the next stationary point \(x=2\), the function starts to increase. After \(x=2\), the function's slope before \(x=3\) might again change its direction, keeping the zero tangents in mind. Thus, between all stationary points from \(x=2,3,4,\) and \(5\), the function exhibits local monotonic behavior due to the horizontal tangents at those points.

Step 4: Construct the Endpoints Behavior

At \(x=6\), the function reaches its maximum value of \(4\). Given that it's the maximum and \(x=5\) is a preceding stationary point, the function generally will move in an increasing trajectory before this point, consistent with its behavior shown by the points before.

Key Concepts

Stationary Points

Stationary points are key features of a function where the derivative is zero, indicating a horizontal tangent line. Consider stationary points like resting spots on a hill: the slope levels out, meaning `no ascent` or `descent`.
For any differentiable function, locating these points helps understand where changes in direction occur. In our exercise, the stationary points at \(x=2, 3, 4, \text{ and } 5\) are places where the function's rate of change becomes temporarily `paused`. This helps in predicting the shape of the graph. Each stationary point provides clues for graph behaviors, helping you visualize the possible humps and dips in the curve.

Maximum and Minimum Values

Maximum and minimum values are specific points where a function reaches its highest or lowest value, within a given domain. Think of these like the peak and valley on a hiker's trail. In your exercise, the function attains a maximum value of 4 at \(x=6\) and a minimum value of \(-2\) at \(x=1\).

• Maximum Value: The highest point, occurring at \(x=6\), means the function 'peaks' here before possibly descending.
• Minimum Value: The lowest point, found at \(x=1\), means it 'troughs' here, potentially climbing upwards thereafter.

Understanding where these extreme values occur guides sketching as these determine the start and end of different directional trends on the graph.

Derivative

The derivative of a function tells us about the slope or rate of change at any given point on the function's curve. If you imagine a road and its inclines, the derivative is like your car's speed gauge, showing whether you're speeding up or slowing down.
In our scenario, the exercise provides insights about derivative values:

• Zero Derivative: At stationary points \(x=2, 3, 4, 5\), where \(f'(x) = 0\), the function's curve stops getting steeper (positive or negative) temporarily.
• Knowing the derivative helps plot the graph not just by spotting stationary points but understanding where to incline and decline the curve around these spots.

This insight is pivotal for accurately sketching the appropriate `humps and valleys`.

Function Behavior Analysis

Analyzing a function's behavior involves observing how the function acts between and at key points. This covers whether it's increasing, decreasing, or remaining constant over certain intervals. With our function, given at \(x=0\) to \(6\):
- From \(x=0\) to \(1\): The function decreases reaching its minimum at \(-2\).- Post-\(x=1\), until \(x=2\): Increase initiates, perhaps ascending steeply or gently up to the next stationary point.- Behavior between stationary points (\(x=2, 3, 4, 5\)) needs careful examination since many local ups and downs occur within these intervals.- Finally, from \(x=5\) to \(6\): An increasing trend resumes, leading up to the maximum at \(x=6\).
Using function analysis results in a deeper understanding of how the graph moves, helping predict any fluctuating path between specified \(x\) domains.