Problem 22 - Calculus Early Transcendentals: Pearson New International Edition

Question

Determine where the graph of the given function is increasing, decreasing, concave up, and concave down. Then sketch the graph (see Example 4). $$ F(x)=x^{6}-3 x^{4} $$

Step-by-Step Solution

Answer

Increasing on \((-\infty, -\sqrt{2})\) and \((\sqrt{2}, \infty)\), decreasing on \((-\sqrt{2}, \sqrt{2})\). Concave up on \((-\infty, -\sqrt{\frac{6}{5}})\) and \((\sqrt{\frac{6}{5}}, \infty)\), concave down on \((-\sqrt{\frac{6}{5}}, \sqrt{\frac{6}{5}})\).

Step 1: Find the First Derivative

To determine where the function is increasing or decreasing, we start by finding the first derivative of the function. The function is \( F(x) = x^6 - 3x^4 \). The first derivative is obtained by differentiating each term:\[ F'(x) = \frac{d}{dx}(x^6) - \frac{d}{dx}(3x^4) = 6x^5 - 12x^3 \]

Step 2: Identify Critical Points

Critical points occur where the derivative is zero or undefined. The first derivative is \( F'(x) = 6x^5 - 12x^3 \). Set this equal to zero and solve for \( x \):\[ 6x^5 - 12x^3 = 0 \]\[ 6x^3(x^2 - 2) = 0 \]This gives us the critical points at \( x = 0 \) and \( x = \pm \sqrt{2} \).

Step 3: Determine Increasing and Decreasing Intervals

Use the critical points to divide the real number line into intervals and test the sign of \( F'(x) \) in each interval to determine increasing and decreasing behavior:- Test interval \((-\infty, -\sqrt{2})\): Choose \(x = -2\), we have \( F'(x) > 0 \) (Increasing)- Test interval \((-\sqrt{2}, 0)\): Choose \(x = -1\), we have \( F'(x) < 0 \) (Decreasing)- Test interval \((0, \sqrt{2})\): Choose \(x = 1\), we have \( F'(x) < 0 \) (Decreasing)- Test interval \((\sqrt{2}, \infty)\): Choose \(x = 2\), we have \( F'(x) > 0 \) (Increasing)

Step 4: Find the Second Derivative

To determine concavity, find the second derivative of \( F(x) = x^6 - 3x^4 \):\[ F''(x) = \frac{d}{dx}(6x^5 - 12x^3) = 30x^4 - 36x^2 \]Factor \( F''(x) \):\[ 6x^2(5x^2 - 6) \]

Step 5: Identify Inflection Points

Set \( F''(x) = 0 \) to find potential inflection points:\[ 6x^2(5x^2 - 6) = 0 \]This gives us \( x = 0 \) and \( x = \pm \sqrt{\frac{6}{5}} \) as potential inflection points.

Step 6: Determine Concave Up and Concave Down Intervals

Analyze the sign of \( F''(x) \) in the intervals defined by the inflection points:- Test interval \((-\infty, -\sqrt{\frac{6}{5}})\): Choose \(x = -1.5\), we have \( F''(x) > 0 \) (Concave Up)- Test interval \((-\sqrt{\frac{6}{5}}, 0)\): Choose \(x = -0.5\), we have \( F''(x) < 0 \) (Concave Down)- Test interval \((0, \sqrt{\frac{6}{5}})\): Choose \(x = 0.5\), we have \( F''(x) < 0 \) (Concave Down)- Test interval \((\sqrt{\frac{6}{5}}, \infty)\): Choose \(x = 2\), we have \( F''(x) > 0 \) (Concave Up)

Step 7: Sketch the Graph

Using the information from the intervals:- Increasing on \((-\infty, -\sqrt{2}) \) and \((\sqrt{2}, \infty)\).- Decreasing on \((-\sqrt{2}, \sqrt{2})\).- Concave up on \((-\infty, -\sqrt{\frac{6}{5}})\) and \((\sqrt{\frac{6}{5}}, \infty)\).- Concave down on \((-\sqrt{\frac{6}{5}}, \sqrt{\frac{6}{5}})\).Plot these characteristics to sketch the graph visually.

Key Concepts

Graph Sketching

Sketching the graph of a function is one of the most essential skills in analyzing functions. It helps to visually interpret the behavior of a function across different intervals. To sketch a graph effectively, understanding the key features like critical points, inflection points, and intervals of increase and decrease is essential.

Here’s a step-by-step guide to graph sketching for the function \( F(x) = x^6 - 3x^4 \):

• **Identify Critical Points and Intervals**: Determine where the function is increasing, decreasing, concave up, and concave down.
• **Plot Key Points**: Use these points to mark where changes in direction or concavity occur.
• **Smooth Connections**: Draw smooth curves that reflect these characteristics, especially around critical and inflection points to ensure the graph accurately represents changes in behavior.

Following these steps helps in constructing a meaningful sketch of the function’s graph.

Critical Points

Critical points are vital in understanding a function's behavior. They occur where the first derivative of a function is zero or undefined. For our function \( F(x) = x^6 - 3x^4 \), we found the critical points by setting the first derivative, \( F'(x) = 6x^5 - 12x^3 \), equal to zero.

Finding critical points involves these steps:

• **Solve \( F'(x) = 0 \)**: Factor the equation to reveal the points where the derivative is zero. For example, \( 6x^3(x^2 - 2) = 0 \) gives critical points at \( x = 0 \) and \( x = \pm \sqrt{2} \).
• **Consider Domain Restrictions**: Ensure the points are within the domain of the original function.

Critical points help to determine where the graph of the function changes from increasing to decreasing or vice versa.

Increasing Intervals

The intervals on which a function is increasing are determined using the first derivative. A function increases when its derivative is positive. For \( F(x) = x^6 - 3x^4 \), the first derivative \( F'(x) = 6x^5 - 12x^3 \) is used to test intervals created by dividing the number line at critical points.

Here's how to determine these intervals:

• **Select a Test Point in Each Interval**: Choose values in intervals defined by critical points ((\(-\infty, -\sqrt{2}\)), (\(-\sqrt{2}, 0\)), (\(0, \sqrt{2}\)), and (\(\sqrt{2}, \infty\))).
• **Evaluate Sign of \( F'(x) \)**: A positive value indicates that the function is increasing in that interval.

From the tests, we see that \( F(x) \) is increasing on the intervals \((\(-\infty, -\sqrt{2}\))\) and \((\(\sqrt{2}, \infty\))\). Recognizing these intervals is crucial for sketching and analyzing a graph effectively.

Concavity Analysis

Concavity refers to the direction and bend of a graph—either upwards or downwards. To determine this, we use the second derivative of the function. For \( F(x) = x^6 - 3x^4 \), the second derivative is \( F''(x) = 30x^4 - 36x^2 \).

Analyze concavity through these steps:

• **Solve \( F''(x) = 0 \)**: Find inflection points where concavity may change, such as \( x = 0 \) and \( x = \pm \sqrt{\frac{6}{5}} \).
• **Test Intervals Between Inflection Points**: Choose values within these intervals to test \( F''(x) \).
• **Determine Concavity**: If \( F''(x) > 0 \), the graph is concave up; if \( F''(x) < 0 \), it is concave down.

By checking concavity, we find that the function is concave up on intervals \((\(-\infty, -\sqrt{\frac{6}{5}}\))\) and \((\(\sqrt{\frac{6}{5}}, \infty\))\). This analysis is key when sketching the graph to enhance the representation of its behavior.