Problem 60 - Calculus of a Single Variable

Question

Inflection Point Consider the function \(f(x)=\sqrt[3]{x}\) . (a) Graph the function and identify the inflection point. (b) Does \(f^{\prime \prime}(x)\) exist at the inflection point? Explain.

Step-by-Step Solution

Answer

The function \(f(x) = \sqrt[3]{x}\) has an inflection point at \(x = 0\), but the second derivative \(f''(0)\) does not exist at this point as it results in a division by zero.

Step 1: Understanding the Function

The function given is \( f(x) = \sqrt[3]{x} \), also known as the cube root function. The graph of this function should show that it's increasing throughout its domain and it has an inflection point at \(x = 0\).

Step 2: Finding the Inflection Point

An inflection point occurs where a function changes concavity. From the way the cube root function behaves, it can be concluded that the inflection point for \( f(x) = \sqrt[3]{x} \) is at \(x = 0\).

Step 3: Computing the Second Derivative

Next, find the second derivative of \( f(x) \). The first derivative of \( f(x) \) is \( f'(x) = \frac{1}{3x^{2/3}} \). The second derivative \( f''(x) \) then is the derivative of \( f'(x) \), that is \( f''(x) = -\frac{2}{9x^{5/3}} \).

Step 4: Evaluating the Second Derivative at the Inflection Point

It's now important to determine whether the second derivative exists at the inflection point \(x = 0\). Substituting \(x = 0\) into \( f''(x) = -\frac{2}{9x^{5/3}} \), it comes clear that \( f''(0) \) is undefined, as you'd be attempting to divide by zero.

Key Concepts

Cube Root Function

The cube root function, defined by the expression \( f(x) = \sqrt[3]{x} \) for each real number x, plays a fundamental role in understanding the behavior of functions and their inflection points.

Like its counterpart the square root function, which produces a 'half parabola' shape on a graph for non-negative values, the cube root function extends this idea to encompass all real numbers, both positive and negative. What is unique about the cube root function is its symmetry. This symmetry is due to the function being odd, meaning it satisfies the condition that \( f(-x) = -f(x) \).

When graphed, the function gradually increases in the positive direction and decreases in the negative direction, intersecting the origin (0,0). This point of intersection is not just a root of the function but, as we'll discuss in the following sections, serves as an inflection point where the concavity of the function changes.

Concavity of Functions

Concavity refers to the curvature of a graphed function and is a critical aspect in determining its shape and behavior. When facing upwards, a function is considered concave up, and when facing downwards, it is concave down. Think of the concavity as the direction a function would bend if it were a flexible wire.

To identify sections of a graph where the function is concave up or down, we look for changes in the slope of the tangent line, or more precisely, the rate at which the derivative of the function is changing. These intervals are where the second derivative of the function is positive (concave up) or negative (concave down).

Inflection Points and Concavity

Inflection points occur where the concavity changes from up to down or vice versa. At these points, the second derivative is zero or undefined. For the cube root function, \( f(x) = \sqrt[3]{x} \), the inflection point is located at the origin. Here, the function transitions from a decreasing to an increasing rate of change, indicating a shift in concavity.

Second Derivative Test

The second derivative test is a useful tool for analyzing the concavity of a function and locating inflection points. By taking the derivative of the function’s first derivative, the second derivative \( f''(x) \) is found, and it gives us the rate of change of the slope of the original function.

If \( f''(x) > 0 \), the original function \( f(x) \) is concave up in that interval, whereas if \( f''(x) < 0 \), \( f(x) \) will be concave down. For inflection points, we search for where the second derivative is zero or does not exist.

Does the Second Derivative Exist at the Inflection Point?

In the case of the cube root function, the second derivative is \( f''(x) = -\frac{2}{9x^{5/3}} \). At the inflection point, \( x = 0 \), \( f''(x) \) is undefined because we cannot divide by zero. Therefore, while \( f(x) \) has an inflection point at the origin, the second derivative test cannot be applied in the conventional sense, since \( f''(0) \) does not exist. This underlines an essential point: the existence of an inflection point is not guaranteed by the existence of a second derivative but rather indicated by a change in concavity that can be observed graphically.