Problem 36 - Calculus Early Transcendentals

Question

Give a graph of the function and identify the locations of all critical points and inflection points. Check your work with a graphing utility. $$ 5 x^{2 / 3}+x^{5 / 3} $$

Step-by-Step Solution

Answer

Critical points: \( x = -2, 0 \). Inflection point: \( x = 1 \).

Step 1: Define the Function

The function given is \( f(x) = 5x^{2/3} + x^{5/3} \). Our goal is to find the critical points and inflection points, then sketch the graph.

Step 2: Find the First Derivative

Calculate the first derivative to locate the critical points. The first derivative is given by applying the power rule to each term:\[ f'(x) = \frac{d}{dx}(5x^{2/3}) + \frac{d}{dx}(x^{5/3}) \]Using the power rule, we have:\[ f'(x) = 5 \cdot \frac{2}{3}x^{-1/3} + \frac{5}{3}x^{2/3} = \frac{10}{3}x^{-1/3} + \frac{5}{3}x^{2/3} \]

Step 3: Find Critical Points

Critical points occur where the first derivative equals zero or is undefined. Set the first derivative to zero:\[ \frac{10}{3}x^{-1/3} + \frac{5}{3}x^{2/3} = 0 \]Multiply through by \( x^{1/3} \) to clear the fraction, yielding:\[ 10 + 5x = 0 \]Solve for \( x \):\[ 5x = -10 \]\[ x = -2 \]Check for any points where the first derivative is undefined, noting that the derivative becomes undefined at \( x = 0 \). Thus, the critical points are \( x = -2 \) and \( x = 0 \).

Step 4: Find the Second Derivative

To find the inflection points, calculate the second derivative \( f''(x) \):\[ f''(x) = \frac{d}{dx}\left(\frac{10}{3}x^{-1/3} + \frac{5}{3}x^{2/3}\right) \]\[ f''(x) = \frac{10}{3} \cdot \left(-\frac{1}{3}\right)x^{-4/3} + \frac{5}{3} \cdot \frac{2}{3}x^{-1/3} \]\[ f''(x) = -\frac{10}{9}x^{-4/3} + \frac{10}{9}x^{-1/3} \]

Step 5: Find Inflection Points

Set \( f''(x) = 0 \) to find the inflection points:\[ -\frac{10}{9}x^{-4/3} + \frac{10}{9}x^{-1/3} = 0 \]Multiply by \( x^{4/3} \) to clear the negative exponents:\[ -\frac{10}{9} + \frac{10}{9}x = 0 \]Solve for \( x \):\[ \frac{10}{9}x = \frac{10}{9} \]\[ x = 1 \]Thus, the inflection point is at \( x = 1 \).

Step 6: Graph the Function

Using the critical points \( x = -2 \) and \( x = 0 \), and the inflection point \( x = 1 \), sketch the graph of \( f(x) = 5x^{2/3} + x^{5/3} \). Observe the behavior of the graph around these key points to ensure it corresponds with what is expected from the calculated derivatives.

Step 7: Verify with a Graphing Utility

Plot the function using a graphing calculator or graphing software to visually confirm the locations of critical points \( x = -2 \), \( x = 0 \), and the inflection point \( x = 1 \). Ensure the graph matches expectations from the calculus analysis.

Key Concepts

Calculus Graphing

When graphing a function in calculus, the primary goal is to interpret the behavior of the function through its derivatives and specific points that define its shape. To graph the function given in this exercise, which is \( f(x) = 5x^{2/3} + x^{5/3} \), we follow several steps:

1. **Identify Key Features**: These are the critical points where the derivative is zero or undefined and the inflection points where the second derivative changes sign. These points provide us with useful information about the function's behavior.
2. **Use Derivatives**: The first and second derivatives help determine where the function increases, decreases, and changes curvature.

By understanding these derivatives, the graph can be accurately sketched to reflect the function's true nature. This process is essential in calculus for visualizing functions, interpreting their behaviors, and solving real-world problems.

Check your work with a graphing utility for accuracy and to ensure that the graph aligns with calculated features such as critical and inflection points.

Inflection Points

Inflection points are critical in understanding the curvature of a function. They are places where the graph of the function changes curvature, meaning it shifts from concave up to concave down, or vice versa.

To find inflection points, the second derivative must be calculated. In this exercise, for the function \( f(x) = 5x^{2/3} + x^{5/3} \), the second derivative is:
\[ f''(x) = -\frac{10}{9}x^{-4/3} + \frac{10}{9}x^{-1/3} \]
Setting \( f''(x) = 0 \) provides the inflection point at \( x = 1 \). This indicates that the graph changes curvature at \( x = 1 \).

Understanding where and how a graph curves offers insights into the function's responsiveness and trend, useful in fields ranging from economics to physics for predicting critical changes in trends or behaviors.

First Derivative Analysis

The first derivative of a function is a valuable tool for determining critical points, which are points where the function's slope is zero or undefined. These points often mark places where the graph reaches a local maximum, minimum, or a saddle point.

For the given function \( f(x) = 5x^{2/3} + x^{5/3} \), the first derivative is found by differentiating each term:
\[ f'(x) = \frac{10}{3}x^{-1/3} + \frac{5}{3}x^{2/3} \]
To find where the slope is zero, solve the equation:
\[ \frac{10}{3}x^{-1/3} + \frac{5}{3}x^{2/3} = 0 \]
The critical points are found at \( x = -2 \) and \( x = 0 \). These points are essential for sketching the graph and understanding intervals where the function might increase or decrease.

In practice, first derivative analysis is fundamental in optimization problems, helping identify points such as profit maximization or cost minimization.