Problem 21 - Calculus Early Transcendentals

Question

Designing a function Sketch a graph of a function \(f\) continuous on \([0,4]\) satisfying the given properties. \(f^{\prime}(1)\) and \(f^{\prime}(3)\) are undefined; \(f^{\prime}(2)=0 ; f\) has a local maximum at \(x=1 ; f\) has a local minimum at \(x=2 ; f\) has an absolute maximum at \(x=3 ;\) and \(f\) has an absolute minimum at \(x=4\).

Step-by-Step Solution

Answer

Question: Sketch the graph of a continuous function \(f\) defined on the interval \([0,4]\) with the given properties: \(f'(1)\) is undefined and \(f'(3)\) is undefined, \(f(1)\) is a local maximum, \(f(2)\) is a local minimum, \(f(3)\) is an absolute maximum, and \(f(4)\) is an absolute minimum. Answer: The graph of the continuous function \(f\) is a piecewise curve with a cusp at \(x=1\), a smooth curve with a local minimum at \(x=2\), a cusp at \(x=3\), and possibly an undefined derivative at \(x=4\). The function changes direction at each critical point: it increases before \(x=1\), decreases between \(x=1\) and \(x=2\), increases between \(x=2\) and \(x=3\), and decreases after \(x=3\).

Step 1: Identifying critical points and their types

We read from the problem statement the critical points, their types, and the information about where the function's first derivative is undefined. The critical points are: 1. \(x=1\): Local maximum and the derivative is undefined 2. \(x=2\): Local minimum, the derivative is zero 3. \(x=3\): Absolute maximum and the derivative is undefined 4. \(x=4\): Absolute minimum

Step 2: Sketching the function using critical points

Now, we are going to sketch the graph of the function using the information about the critical points: 1. At \(x=1\), there is a local maximum and the derivative is undefined, so we draw a cusp (a sharp point) at this point and the function will change from increasing to decreasing. 2. At \(x=2\), the function has a local minimum and the derivative is zero, so we draw a smooth curve with a local minimum at this point and the function will change from decreasing to increasing. 3. At \(x=3\), there is an absolute maximum and the derivative is undefined, so we draw a cusp (a sharp point) at this point and the function will change from increasing to decreasing. 4. At \(x=4\), the function has an absolute minimum, and as \(f'(3)\) is undefined and \(f'(1)\) is undefined, it is likely that the function is also undefined at this point. After sketching the graph, we can see that the function is in the form of a continuous piecewise function, where each interval between critical points has a unique form that fits the necessary properties.

Key Concepts

Function Sketching

Function sketching involves creating a visual representation of a mathematical function. This is particularly useful to understand the behavior of the function over a specific interval.
To sketch a function, it is essential to know certain properties such as critical points, local maxima and minima, and any undefined points of the derivative. These properties provide the necessary landmarks to shape the sketch correctly.

• **Critical Points:** Points where the derivative is zero or does not exist.
• **Local Extrema:** Points where the function reaches a local maximum or minimum.
• **Shape & Continuity:** Ensure the graph is continuous (no breaks) over the defined interval.

For the function in our exercise, we identify critical points at specific x-values and use them to plot the graph. The knowledge of where the function increases or decreases and where it potentially has cusps or smooth changes helps construct a comprehensive sketch.

Derivatives

A derivative represents the rate of change of a function with respect to a variable. It is a fundamental tool in calculus for identifying the slope of the function at any given point.
In this exercise, the derivative helps identify key behavior of the function:

• **Behavior at Critical Points:** The sign and value of the derivative at critical points tell us how the function behaves around these points.
• **Undefined Derivative:** In situations like cusps, the derivative does not exist, indicating a change that is not smooth.

The specific derivative information given is:

• At x=1 and x=3, derivatives are undefined, indicating sharp points.
• At x=2, the derivative is zero, marking a smooth bottom curve.

Understanding where the derivative is undefined or zero is essential for sketching the function accurately, as these points often indicate local maxima, minima, or points of inflection.

Critical Points

Critical points are values of x where a function's derivative is zero or undefined. These points are crucial for identifying potential local maxima or minima in the function.
For our exercise, the critical points are:

• **x=1:** The derivative is undefined, and there is a local maximum. This suggests a cusp or sharp peak in the graph.
• **x=2:** The derivative equals zero here, so it's a local minimum with a smooth shape.
• **x=3:** Again, the derivative is undefined, indicating an absolute maximum with a cusp.
• **x=4:** While not needing a derivative to identify, this point is the absolute minimum.

In graphing, these points structure the function's graph and indicate where changes in direction take place. Identifying and understanding these points is essential to accurately depict the function's behavior.

Local Maxima and Minima

Local maxima and minima refer to points on a graph where the function reaches its highest or lowest value, respectively, within a certain interval.
These points provide significant insight into the function’s pattern:

• **Local Maximum:** This is where the function value is higher than neighboring points. In the example, x=1 and x=3 are local maxima. The graph has peaks at these x-values.
• **Local Minimum:** This is where a function's value is lower than its surrounding points. In this function, x=2 is a local minimum, depicted as a valley in the graph.

A thorough understanding of local maxima and minima allows one to determine important features of the function's shape and is fundamental in predicting and understanding function behavior over intervals.