Problem 23 - Precalculus with Calculus Previews

Question

In Problems \(23-44\), proceed as in Example 2 and sketch the graph of the given polynomial function \(f\). \(f(x)=x^{3}-4 x\)

Step-by-Step Solution

Answer

The graph of \(f(x) = x^3 - 4x\) has critical points at \(x = \pm\sqrt{\frac{4}{3}}\), increasing in \((\sqrt{\frac{4}{3}}, \infty)\) and \((-\infty, -\sqrt{\frac{4}{3}})\), decreasing in \((-\sqrt{\frac{4}{3}}, \sqrt{\frac{4}{3}})\), and an inflection point at \(x = 0\).

Step 1: Determine Critical Points

To find the critical points of the function, we first need to find the derivative of the function \(f(x) = x^3 - 4x\). The derivative is \(f'(x) = 3x^2 - 4\). Set \(f'(x) = 0\) to find critical points: \(3x^2 - 4 = 0\). Solving this equation gives \(x = \pm\sqrt{\frac{4}{3}}\).

Step 2: Determine Intervals of Increase and Decrease

Using the critical points, divide the number line into intervals: \((-\infty, -\sqrt{\frac{4}{3}})\), \((-\sqrt{\frac{4}{3}}, \sqrt{\frac{4}{3}})\), and \((\sqrt{\frac{4}{3}}, \infty)\). Test points in these intervals to determine if the function is increasing or decreasing. For example, test \(x = -2\) in \(-\infty, -\sqrt{\frac{4}{3}})\), \(x = 0\) in \((-\sqrt{\frac{4}{3}}, \sqrt{\frac{4}{3}})\), and \(x = 2\) in \((\sqrt{\frac{4}{3}}, \infty)\). This analysis can show whether the derivative is positive (increasing) or negative (decreasing).

Step 3: Determine Inflection Points

Find the second derivative \(f''(x)\) which is \(6x\). Set \(f''(x) = 0\) to find potential inflection points: \(6x = 0\) yielding \(x = 0\). Check the sign change in \(f''(x)\) around \(x = 0\) to confirm it's an inflection point.

Step 4: Sketch the Graph

Using information from Steps 1-3, sketch the graph. The critical points and intervals show where the graph peaks or dips and where it is increasing or decreasing. The inflection point gives a changing concavity. The graph should show symmetry around the origin consistent with an odd-degree polynomial (cubic function) and have roots where \(f(x) = 0\), which can be found by solving \(x^3 - 4x = 0\) (use the quadratic formula if needed).

Key Concepts

Critical Points

Critical points are vital in understanding the shape and behavior of a graph, particularly for polynomial functions. They are where the first derivative of a function is zero or undefined.
For the function, \(f(x) = x^3 - 4x\), we first calculate the derivative, \(f'(x) = 3x^2 - 4\). By setting the derivative to zero, \(3x^2 - 4 = 0\), we solve for \(x\), to find the critical points: \(x = \pm\sqrt{\frac{4}{3}}\).
These points can signify a local maximum, minimum, or saddle point. They are locations where the function stops increasing or decreasing momentarily.

Derivative

The derivative of a function is the mathematical tool for finding the rate at which the function's value changes. It's crucial for determining where the function increases or decreases.
In our cubic function example, the derivative is \(f'(x) = 3x^2 - 4\). By plugging numbers into this derivative, we can test intervals around critical points. It helps us determine whether the function is increasing or decreasing in those ranges.
For instance, if you test a value within an interval, like \(x = 0\) between \(-\sqrt{\frac{4}{3}}\) and \(\sqrt{\frac{4}{3}}\), you will find the derivative is negative. This indicates that the function is decreasing in this region.

Inflection Points

Inflection points are where a function changes its curvature from concave up to concave down or vice versa. They are found by examining the second derivative of the function.
For the function \(f(x) = x^3 - 4x\), the second derivative is \(f''(x) = 6x\). By setting it to zero, \(6x = 0\), we solve for potential inflection points, giving \(x = 0\).
To confirm this, check the sign change around \(x = 0\) in \(f''(x)\). If the second derivative changes sign as you pass through \(x = 0\), it confirms an inflection point, indicating a shift in concavity.

Cubic Function

A cubic function is a type of polynomial of degree three. These functions can have up to three real roots and are known for their signature 'S' shape on a graph.
The general form of a cubic function is \(f(x) = ax^3 + bx^2 + cx + d\). In our example, the function is \(f(x) = x^3 - 4x\).
Cubic functions are essential because of their versatility in modeling real-world scenarios. They can show a point of inflection and have both increasing and decreasing intervals. For this reason, understanding their derivatives and critical points is integral for sketching accurate graphs and predicting behavior over certain intervals.