Question

Use the derivative \(f^{\prime}\) to determine the \(x\) -coordinates of the local maxima and minima of \(f\), and the intervals on which \(f\) is increasing or decreasing. Sketch a possible graph of \(f(f\) is not unique). $$f^{\prime}(x)=(x-1)(x+2)(x+4)$$

Short answer

Based on the analysis, sketch a rough graph of the function f(x) and label the critical points, intervals of increase and decrease, and local maxima and minima. Be sure to include the local maximum at \(x=-4\), the local minimum at \(x=-2\), and the neither maximum nor minimum at \(x=1\). Also, indicate the intervals of increase and decrease on the graph.

Step-by-step solution

Determine Critical Points

To find the critical points, we need to find the x-values where \(f'(x) = 0\). Since our derivative is already factored, we can easily see the critical points: $$ f^{\prime}(x) = (x-1)(x+2)(x+4) = 0 $$ The critical points occur at \(x = 1, -2, -4\).

Classify the Critical Points

Now, we can use the first derivative test to determine which of these critical points are maximums or minimums. The idea is to test the sign of the first derivative in each interval cut by the critical points. If \(f'(x) > 0\) in an interval, the function is increasing there. If \(f'(x) < 0\), the function is decreasing. We have the intervals: - \((-\infty, -4)\) - \((-4, -2)\) - \((-2, 1)\) - \((1, \infty)\) Testing a value within each interval: - \(f'(-5) = (-6)(-3)(-1) > 0\), so \(f(x)\) is increasing. - \(f'(-3) = (-4)(-1)(1) < 0\), so \(f(x)\) is decreasing. - \(f'(0) = (-1)(2)(4) > 0\), so \(f(x)\) is increasing. - \(f'(2) = (1)(4)(6) > 0\), so \(f(x)\) is increasing.

Intervals of Increase/Decrease and Local Maxima/Minima

Based on the results from the first derivative test, we can conclude the following: Intervals of increase: \((-\infty, -4) \cup (-2, \infty)\) Intervals of decrease: \((-4, -2)\) Since the function is increasing to the left of \(x = -4\) and decreasing to the right, we have a local maximum at \(x = -4\). Since the function is decreasing to the left of \(x = -2\) and increasing to the right, we have a local minimum at \(x = -2\). Since the function is increasing to the left and right of \(x = 1\), this critical point is neither a maximum nor minimum.

Sketch a Possible Graph of f(x)

With the information gathered from the previous steps, sketch a possible graph of \(f(x)\). Keep in mind that \(f(x)\) is not unique, so our sketch is just one possible solution. Since we don't have the expression for \(f(x)\), we cannot determine specific y-values at local maximum and minimum points. However, we can still create a rough sketch by placing the local maximum and minimum points at the x-values determined, and making sure the function follows the increasing and decreasing intervals. - Place a local maximum at \(x=-4\). - Place a local minimum at \(x=-2\). - Make sure the function is increasing from \(-\infty\) to \(-4\), decreasing from \(-4\) to \(-2\), and increasing from \(-2\) to \(\infty\). Remember to label the critical points, intervals of increase and decrease, and local maxima and minima.

Key concepts

The First Derivative Test

The First Derivative Test is an essential tool for determining the local maxima and minima of functions.

Here's how it works: First, you calculate the derivative of the function, which gives the rate of change. If the derivative is zero at a particular point, that point is called a critical point. To figure out whether these critical points are local maxima, minima, or neither, you test the sign of the derivative just before and after the point.

• If the sign changes from positive to negative, the function is increasing and then decreasing, which indicates a local maximum.
• If the sign changes from negative to positive, it's a local minimum.
• If there's no sign change, the critical point is neither a maximum nor minimum.

In the exercise, the derivative function equals zero at three different x-values, leading to three critical points to examine with this test.

Identifying Local Maxima and Minima

Local maxima and minima play a significant role in understanding the behavior of functions.

A local maximum is where the function's value is higher than all nearby points, and a local minimum is where it's lower. These points provide valuable insights into the function's graph and are often where interesting things happen, like the peak of a hill (maximum) or the bottom of a valley (minimum). We use the First Derivative Test to pinpoint these features by observing the critical points.

Referring to the solution steps, we noticed a local maximum at the critical point where the function switched from increasing to decreasing, and a local minimum where it switched back to increasing. These critical points help reveal the function's landscape and shape.

Increasing and Decreasing Intervals

The intervals where a function increases or decreases are crucial for sketching its graph.

A function is increasing on an interval if its slope (derivative) is positive throughout that interval; similarly, it's decreasing if the slope is negative. By determining where the derivative of the function is positive or negative, we can chart the function's 'ups and downs' across its domain. These intervals tell us which way the function is trending at different sections.

In the provided exercise, we saw that the function was increasing over two separate intervals and decreasing over another. By understanding these patterns, we can plot the behavior of the function (e.g., upward or downward) between the critical points, better visualizing the entire graph.