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Chapter 4: Problem 8

Sketch the graph of a function \(f\) that has a local minimum value at a point \(c\) where \(f^{\prime}(c)\) is undefined.

Short Answer

Expert verified
Question: Sketch a graph of a function that has a local minimum at a point where its derivative is undefined. Describe the properties of the function and its graph. Answer: A function that has a local minimum at a point where its derivative is undefined is \(f(x) = |x^3|\). This function has a cusp at x = 0, which corresponds to a local minimum. It is continuous, and its derivative is \(3x^2\) for \(x > 0\) and \(-3x^2\) for \(x < 0\). However, the derivative is undefined at \(x = 0\). The graph of this function has a cusp at the origin \((0, 0)\) and is symmetric with respect to the y-axis.

Step by step solution

01

Create an example function

An example of a function that satisfies the given condition is \(f(x) = |x^3|\) , which has a cusp at x = 0. Now, we will find where the local minimum for this function occurs and will confirm that its derivative is undefined at that point.
02

Find the local minimum

To find the local minimum, we have to analyze the function and inspect its turning points. For \(f(x) = |x^3|\), we observe that it has a cusp at \(x = 0\), and since this cusp is the lowest point for the function, it corresponds to a local minimum.
03

Check the continuity of the function

For the point to be a local minimum, the function should also be continuous at that point. \(|x^3|\) is continuous everywhere, as an absolute value of any polynomial is continuous. Hence, \(f(x)\) is continuous at \(x = 0\).
04

Examine the derivative

We will now find the derivative of the function to explore its properties. Since \(f(x) = |x^3|\), we have to take the derivative separately for positive and negative values of \(x\). For \(x > 0\), \(f(x) = x^3\), and for \(x < 0\), \(f(x) = -x^3\). For \(x > 0\), \(f^{\prime}(x) = 3x^2\) and for \(x < 0\), \(f^{\prime}(x) = -3x^2\). However, at \(x = 0\), the derivative does not exist since there is a cusp.
05

Sketch the graph

Now that we know that \(f(x) = |x^3|\) has a local minimum at \(x = 0\) and the derivative is undefined at that point, we can sketch the graph. The graph will have a cusp at \(x = 0\) and will be increasing on both sides of the cusp. Using this information, we can now sketch the graph of the function \(f(x) = |x^3|\). It will have a cusp at the origin \((0, 0)\) and will be symmetric with respect to the y-axis.

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