Problem 13 - Calculus with Concepts in Calculus

Question

Find the intervals on which the graph of the function is concave upward and those on which it is concave downward. Then sketch the graph of the function. $$ g(x)=x^{4}-4 x $$

Step-by-Step Solution

Answer

The graph is concave upward everywhere and there are no intervals where it's concave downward.

Step 1: Find the Second Derivative

To determine concavity, we first need to find the second derivative of the function. Start with the first derivative of \( g(x) = x^4 - 4x \). The first derivative is \( g'(x) = 4x^3 - 4 \). Now take the derivative of \( g'(x) \) to get the second derivative: \( g''(x) = 12x^2 \).

Step 2: Identify Critical Points of the Second Derivative

Set \( g''(x) = 12x^2 \) equal to zero to find the critical points where concavity may change. Solve the equation \( 12x^2 = 0 \), which gives \( x = 0 \). Thus, \( x = 0 \) is a potential point of inflection.

Step 3: Test the Intervals Around the Critical Points

Determine the sign of \( g''(x) \) around \( x = 0 \) to identify concavity intervals. For \( x < 0 \), choose \( x = -1 \): \( g''(-1) = 12(-1)^2 = 12 \), which is positive. For \( x > 0 \), choose \( x = 1 \): \( g''(1) = 12(1)^2 = 12 \), which is also positive. Therefore, the graph is concave upward on both intervals.

Step 4: Conclusion on Concavity

Since \( g''(x) = 12x^2 \) is positive for all \( x \), the function is concave upward everywhere. There are no intervals where the function is concave downward.

Step 5: Sketch the Graph

Draw the graph of the function \( g(x) = x^4 - 4x \). Since it is concave upward on the entire domain, it will look like a parabola opening upwards. Plot a few points to guide the shape, particularly around \( x = 0 \) and analyze the root \( x \) values to get an accurate graph.

Key Concepts

Second Derivative

The second derivative is key to understanding the concavity of a function. It is calculated by taking the derivative of the first derivative of a function. For instance, starting with the function given in the exercise, \( g(x) = x^4 - 4x \), we first find the first derivative: \( g'(x) = 4x^3 - 4 \). Then, by deriving again, we obtain the second derivative: \( g''(x) = 12x^2 \). The second derivative tells us how the rate of change (slope) of the function is itself changing. If the second derivative is positive over an interval, the function's graph will be concave upward in that region. Conversely, if the second derivative is negative, the graph is concave downward.

Concavity Test

The concavity test is a simple method using the sign of the second derivative to determine whether a function is concave up or down on specific intervals. To perform this test, follow these steps:

• Calculate the second derivative, as explained earlier.
• Identify critical points where the second derivative equals zero or is undefined. These points are potential inflection points where the concavity could change.
• Check intervals around those critical points to see where the second derivative is positive or negative.

In the provided exercise, the second derivative \( g''(x) = 12x^2 \) is zero at \( x = 0 \). By testing intervals around this critical point, we found that the second derivative remains positive for all \( x \), indicating that the graph is concave upward everywhere.

Critical Points

Critical points are significant in analyzing a function as they can indicate points where the function changes behavior, for example where it moves from concave up to concave down or vice versa. In the context of concavity, critical points of the second derivative help us find potential inflection points.Here's how you deal with critical points for concavity:

• Set the second derivative equal to zero: \( g''(x) = 0 \).
• Solve this equation to find critical points. In our exercise, \( 12x^2 = 0 \) gives \( x = 0 \).
• Analyze the behavior of the second derivative around these critical points to determine where the graph is concave upward or downward.

In this scenario, since there was no change in the sign of the second derivative around \( x = 0 \), there were no inflection points, and the graph remained concave upward.

Function Sketching

Function sketching is an excellent way to visualize the behavior of a function, especially after determining its concavity from the second derivative. To sketch a function like \( g(x) = x^4 - 4x \), consider:

• Finding the function's roots by solving \( g(x) = 0 \), which helps locate where the graph crosses the x-axis.
• Understanding symmetry and intercepts that may help in sketching symmetric functions or locating y-intercepts.
• Using the previous concavity analysis, which revealed the graph is concave upward everywhere. This means the general shape is a smooth curve opening upwards.
• Plotting several points for precision, especially around critical points or changes in direction.

The ultimate goal is to draw a graph that accurately represents the function's behavior as prescribed by the calculus insights gathered from derivatives.