Problem 32 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Determine where the function is concave upward and where it is concave downward. $$ f(x)=\sqrt[3]{x} $$

Step-by-Step Solution

Answer

The function \(f(x) = \sqrt[3]{x}\) is concave upward when \(x < 0\) and concave downward when \(x > 0\).

Step 1: Find the first derivative of the function

The given function is \(f(x) = \sqrt[3]{x}\). We can rewrite it as \(f(x) = x^{\frac{1}{3}}\). To find the first derivative, we'll differentiate the function. Using the power rule: \(f'(x) = \frac{1}{3}x^{\frac{1}{3}-1}\) Which simplifies to: \(f'(x) = \frac{1}{3}x^{-\frac{2}{3}}\)

Step 2: Find the second derivative of the function

Now, to find the concavity of the function, we need to find the second derivative. Differentiate the first derivative \(f'(x)\): \(f''(x) = -\frac{2}{9}x^{-\frac{5}{3}}\)

Step 3: Determine the concavity of the function

The second derivative \(f''(x) = -\frac{2}{9}x^{-\frac{5}{3}}\) helps us determine the concavity. 1. If \(f''(x) > 0\), the function is concave upward. 2. If \(f''(x) < 0\), the function is concave downward. Since the second derivative has a negative coefficient \(-\frac{2}{9}\), and any \(x\) value in the exponent is also negative, this means: 1. When \(x > 0\), the exponent is negative, making the second derivative negative, so the function is concave downward. 2. When \(x < 0\), the exponent is negative, making the second derivative positive, so the function is concave upward. Thus, the function is concave upward when \(x < 0\) and concave downward when \(x > 0\).

Key Concepts

First Derivative

The first derivative of a function gives us valuable information about the behavior of the function in terms of its rate of change. In our exercise, we're working with the function \(f(x) = \sqrt[3]{x}\), which can be rewritten as \(f(x) = x^{\frac{1}{3}}\). To determine how this function increases or decreases, we compute the first derivative using differentiation.

Applying the power rule, \(f'(x) = \frac{1}{3}x^{\frac{1}{3}-1}\), simplifies to \(f'(x) = \frac{1}{3}x^{-\frac{2}{3}}\). This derivative essentially tells us how steep the function is at any given point on the \(x\)-axis. Notice the negative exponent, which will affect the shape and direction of the graph as you move along the \(x\)-axis.

• Positive first derivative implies the function is increasing.
• Negative first derivative implies the function is decreasing.

Understanding the first derivative is the first step in analyzing the concavity of functions.

Second Derivative

The second derivative provides insight into the concavity of a function, which essentially tells us how the function curves. After finding the first derivative \(f'(x) = \frac{1}{3}x^{-\frac{2}{3}}\), we proceed to find the second derivative.

Applying differentiation again, the second derivative becomes \(f''(x) = -\frac{2}{9}x^{-\frac{5}{3}}\). This expression encompasses more information about the rate at which the slope \(f'(x)\) is changing.

• Positive second derivative: The function is concave upward.
• Negative second derivative: The function is concave downward.

In this exercise, the second derivative has a consistent negative factor of \(-\frac{2}{9}\), shedding light on the curvature of the function across different regions of \(x\).

Power Rule

The power rule is an essential tool in calculus for differentiating functions of the form \(x^n\). It states that the derivative of \(x^n\) is \(nx^{n-1}\), a simple yet powerful method used frequently.

For our function \(f(x) = x^{\frac{1}{3}}\), applying the power rule gives us \( \left(\frac{1}{3} \right)x^{\frac{1}{3} - 1} = \frac{1}{3}x^{-\frac{2}{3}}\). The rule helps simplify the differentiation process, allowing us to tackle more complex functions efficiently. Remember:

• Identify the exponent \(n\).
• Multiply by \(n\) and reduce the exponent by one.

This approach is foundational for understanding how derivatives are calculated, playing a crucial role in exploring both first and second derivatives.

Concave Upward and Downward

Concavity describes how a function curves along the \(x\)-axis. In simpler terms, it tells us whether the graph of the function is shaped like a cup or an upside-down cup in specific intervals.

Using the second derivative \(f''(x) = -\frac{2}{9}x^{-\frac{5}{3}}\), we determine:

• If \(f''(x) > 0\), the function is concave upward - similar to an upright cup.
• If \(f''(x) < 0\), the function is concave downward - more like an upside-down cup.

In our exercise, since \(f''(x)\) develops from a consistent negative coefficient, \(f''(x)\) is negative for \(x > 0\) implying concave downward, and positive \(x < 0\) implying concave upward. Recognizing these signs is key to understanding the nature of the function's curvature.