Problem 26 - Applied Calculus

Question

Draw the graph of a continuous function \(y=f(x)\) that satisfies the following three conditions: \- \(f^{\prime}(x)>0\) for \(13\) \- \(f^{\prime}(x)=0\) at \(x=1\) and \(x=3\)

Step-by-Step Solution

Answer

The graph is a curve with a local minimum at \(x=1\) and a local maximum at \(x=3\).

Step 1: Understand Critical Points and Intervals

First, identify where the derivative of the function, \(f'(x)\), is zero or changes sign. The conditions \(f'(x)=0\) at \(x=1\) and \(x=3\) mean these points are critical points. These points usually indicate either a local maximum or minimum.

Step 2: Analyze Interval \(x<1\)

For \(x<1\), we have \(f'(x)<0\), which means that the function is decreasing in this interval. As \(x\) approaches 1 from the left, the function should continue to decrease until \(x = 1\).

Step 3: Analyze Interval \(1<x<3\)

In this interval, we have \(f'(x)>0\), which means that the function is increasing. This implies that there is a change in behavior at \(x=1\) from decreasing to increasing, indicating a local minimum at \(x=1\).

Step 4: Analyze Interval \(x>3\)

For \(x>3\), \(f'(x)<0\), which means the function is decreasing in this interval. Since \(f'(x)=0\) at \(x = 3\) and changes from positive to negative, this indicates a local maximum at \(x=3\).

Step 5: Sketch the Graph

Plot the above behavior on a graph. Start with a decreasing curve approaching \(x = 1\). At \(x=1\), draw a curve that changes to increasing, reaching a peak within the interval \(1

Key Concepts

Derivative

In calculus, the derivative plays a crucial role as it represents the rate of change of a function concerning its input variable. Think of the derivative as the function's fingerprint revealing its behavior at every point along its graph. For any function \( f(x) \), the derivative \( f'(x) \) is the limit of the average rate of change over an interval as the interval approaches zero.
Knowing the derivative helps identify where the function is increasing or decreasing:

• If \( f'(x) > 0 \), the function is increasing.
• If \( f'(x) < 0 \), the function is decreasing.
• If \( f'(x) = 0 \), the function is constant at that point, known as critical points or stationary points.

Understanding derivatives is essential for exploring deeper aspects of functions, like critical points and intervals of increase and decrease, which we will delve into next.

Critical Points

Critical points are points on the graph of a function where the derivative equals zero or is undefined. These points are vital in calculus because they often indicate potential maxima or minima in the function. To identify critical points in a function \( f(x) \), we set its derivative \( f'(x) \) to zero and solve for \( x \).
In our context, the critical points are at \( x=1 \) and \( x=3 \), where \( f'(x)=0 \). These points are interesting because they hint at locations where the function might change from increasing to decreasing, or vice versa, signaling local maxima or minima.
To further investigate a critical point:

• Check the sign of \( f'(x) \) around the point. If it changes from negative to positive, the point is a local minimum.
• If it changes from positive to negative, it is a local maximum.

Analyzing these points will guide us in better understanding the overall shape and behavior of the function.

Decreasing Function

A function is said to be decreasing in an interval if its derivative is negative over that interval. This means the function values descend as \( x \) increases. In simpler words, imagine a slope declining downwards in this domain. In our exercise, the function is decreasing in two regions: for \( x<1 \) and \( x>3 \), due to \( f'(x)<0 \) in these intervals.
When observing the graph, this means:

• Before \( x=1 \), the function approaches from above and decreases until it hits the point \( x=1 \).
• Beyond \( x=3 \), the function descends again, indicating a peak at \( x=3 \).

Recognizing these decreasing intervals helps confirm the location of critical points and the transitions in the function's behavior, aligning with what we expect from \( f'(x) \)'s sign analysis.

Increasing Function

An increasing function means that as \( x \) moves forward, the function's output moves upwards in its graph. Mathematically, this occurs when the derivative \( f'(x) \) is positive across an interval. In our example, the function is increasing between \( x=1 \) and \( x=3 \), indicating a change from a decrease before \( x=1 \) to an increase up to \( x=3 \).
Here's what happens visually:

• The function rises as \( x \) travels from 1 to 3, marking a local minimum starting at \( x=1 \).
• The increase persists right until \( x=3 \), where the function reaches a peak.

These increasing intervals are especially important as they define areas of local minima leading up to points of local maxima. Recognizing where a function increases extends your insight into its graph shape and behavior, ensuring a comprehensive understanding.

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Problem 26

Next

Problem 27

Other exercises in this chapter

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Problem 26

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Problem 27

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Key Concepts

Derivative

In calculus, the derivative plays a crucial role as it represents the rate of change of a function concerning its input variable. Think of the derivative as the function's fingerprint revealing its behavior at every point along its graph. For any function \( f(x) \), the derivative \( f'(x) \) is the limit of the average rate of change over an interval as the interval approaches zero.
Knowing the derivative helps identify where the function is increasing or decreasing:

• If \( f'(x) > 0 \), the function is increasing.
• If \( f'(x) < 0 \), the function is decreasing.
• If \( f'(x) = 0 \), the function is constant at that point, known as critical points or stationary points.

Understanding derivatives is essential for exploring deeper aspects of functions, like critical points and intervals of increase and decrease, which we will delve into next.

Critical Points

Critical points are points on the graph of a function where the derivative equals zero or is undefined. These points are vital in calculus because they often indicate potential maxima or minima in the function. To identify critical points in a function \( f(x) \), we set its derivative \( f'(x) \) to zero and solve for \( x \).
In our context, the critical points are at \( x=1 \) and \( x=3 \), where \( f'(x)=0 \). These points are interesting because they hint at locations where the function might change from increasing to decreasing, or vice versa, signaling local maxima or minima.
To further investigate a critical point:

• Check the sign of \( f'(x) \) around the point. If it changes from negative to positive, the point is a local minimum.
• If it changes from positive to negative, it is a local maximum.

Analyzing these points will guide us in better understanding the overall shape and behavior of the function.

Decreasing Function

A function is said to be decreasing in an interval if its derivative is negative over that interval. This means the function values descend as \( x \) increases. In simpler words, imagine a slope declining downwards in this domain. In our exercise, the function is decreasing in two regions: for \( x<1 \) and \( x>3 \), due to \( f'(x)<0 \) in these intervals.
When observing the graph, this means:

• Before \( x=1 \), the function approaches from above and decreases until it hits the point \( x=1 \).
• Beyond \( x=3 \), the function descends again, indicating a peak at \( x=3 \).

Recognizing these decreasing intervals helps confirm the location of critical points and the transitions in the function's behavior, aligning with what we expect from \( f'(x) \)'s sign analysis.

Increasing Function

An increasing function means that as \( x \) moves forward, the function's output moves upwards in its graph. Mathematically, this occurs when the derivative \( f'(x) \) is positive across an interval. In our example, the function is increasing between \( x=1 \) and \( x=3 \), indicating a change from a decrease before \( x=1 \) to an increase up to \( x=3 \).
Here's what happens visually:

• The function rises as \( x \) travels from 1 to 3, marking a local minimum starting at \( x=1 \).
• The increase persists right until \( x=3 \), where the function reaches a peak.

These increasing intervals are especially important as they define areas of local minima leading up to points of local maxima. Recognizing where a function increases extends your insight into its graph shape and behavior, ensuring a comprehensive understanding.