Question

Functions from derivatives Use the derivative \(f^{\prime}\) to determine the \(x\) -coordinates of the local maxima and minima of \(f,\) and the intervals of increase and decrease. Sketch a possible graph of \(f(f\) is not unique). $$f^{\prime}(x)=(x-1)(x+2)(x+4)$$

Short answer

Question: Identify the local maxima, local minima, and intervals of increase and decrease for the function \(f(x)\) with the derivative \(f'(x) = (x - 1)(x + 2)(x + 4)\). Also, provide a sketch of a possible graph for the function. Answer: The function \(f(x)\) has a local maximum at \(x = -4\), and a local minimum at \(x = -2\). The function increases on the intervals \((-\infty, -4)\), \((-2, 1)\), and \((1, \infty)\), and it decreases on the interval \((-4, -2)\). The sketch of a possible graph of the function should show these key features, but may not be unique due to different initial points or constant shifts.

Step-by-step solution

Find the critical points

To find the critical points of the function, set the derivative equal to zero, and solve for \(x\): $$f'(x) = (x-1)(x+2)(x+4) = 0$$ The critical points occur at \(x = 1, -2, -4\).

Determine the intervals of increase and decrease

We will create a sign chart to determine the intervals of increase and decrease. We place the critical points 1, -2, and -4 in the number line and test numbers from each interval to check if the derivative is positive or negative on each interval: Interval: \((-\infty, -4)\) Test point: \(x = -5\) \(f'(-5) = (-6)(1)(-1) > 0\) (positive) Interval: \((-4, -2)\) Test point: \(x = -3\) \(f'(-3) = (-4)(1)(1) < 0\) (negative) Interval: \((-2, 1)\) Test point: \(x = 0\) \(f'(0) = (1)(-2)(-4) > 0\) (positive) Interval: \((1, \infty)\) Test point: \(x = 2\) \(f'(2) = (1)(4)(6) > 0\) (positive) Based on the sign chart, the function increases on \((-\infty, -4)\), \((-2, 1)\), and \((1, \infty)\), and it decreases on \((-4, -2)\).

Identify local maxima and minima

Now we will identify the local maxima and minima by examining the critical points and their corresponding intervals: - At \(x = -4\), \(f'(x)\) changes from positive to negative, which indicates a local maximum. - At \(x = -2\), \(f'(x)\) changes from negative to positive, which indicates a local minimum. - At \(x = 1\), \(f'(x)\) remains positive, which suggests that there is no local maxima or minima at this point.

Sketch a possible graph of the function

Using the information we gathered from the previous steps, we can sketch a possible graph for the function \(f(x)\): 1. At \(x = -4\), we have a local maximum. 2. At \(x = -2\), we have a local minimum. 3. There are no local maxima or minima at \(x = 1\). 4. The function increases on \((-\infty, -4)\), \((-2, 1)\), and \((1, \infty)\). 5. The function decreases on \((-4, -2)\). Keep in mind that the sketch is not unique, as any function with the same derivative can have a different initial point or a constant shift. Nevertheless, the sketch will need to follow the characteristics we've determined from the analysis of the derivative.

Key concepts

Critical Points

Critical points of a function are the values of the independent variable, usually denoted as 'x', at which the function's derivative is zero or undefined. These points are crucial because they are potential locations for local maxima, minima, or points of inflection, which are key aspects in understanding the behavior of a function.

In the given exercise, to find the critical points, we set the derivative equal to zero: \(f'(x) = (x-1)(x+2)(x+4) = 0\). Solving this equation gives us the critical points: \(x = 1, -2, -4\). These are the exact values where the function 'f' may change its trend from increasing to decreasing or vice versa.

Intervals of Increase and Decrease

Determining the intervals of increase and decrease in a function involves analyzing the sign of its derivative over various intervals. A positive derivative on an interval means that the function is increasing over that interval, while a negative derivative indicates a decrease.

In our exercise, by using test points and creating a sign chart, we've established intervals based on the sign of the derivative: the function is increasing on \( (-\infty, -4) \), \( (-2, 1) \), and \( (1, \infty) \) as shown when substituting test points into \(f'(x)\). Conversely, it is decreasing on \( (-4, -2) \) since the derivative is negative. This analysis helps explain where the function is rising or falling.

Local Maxima and Minima

Local maxima and minima refer to the high and low points within a specific section of a function's graph. A local maximum is a point where the function takes on the largest value in the neighboring interval, whereas a local minimum is where the function takes on the smallest value.

From the derivative \(f'(x)\), we've identified a local maximum at \(x = -4\) because the derivative changes from positive to negative, indicating a peak. Similarly, a local minimum is present at \(x = -2\) because the derivative changes from negative to positive, indicating a trough. As for \(x = 1\), no local extremum is noted since the derivative doesn't change sign.

Sketching Graphs of Functions

Sketching the graph of a function using its derivative involves locating the critical points and determining where the function is increasing and decreasing. It is also necessary to check for constant areas where the derivative is zero over an interval.

In the solution to our exercise, after identifying critical points, intervals of increase and decrease, and local maxima and minima, we sketch a graph reflecting these characteristics: a rising curve up to \(x = -4\), a fall until \(x = -2\), a rise again past \(x = 1\), and continuing upwards. It's important to remember that while the graph of \(f(x)\) isn't unique due to possible vertical shifts, it must honor the determined characteristics given by the derivative information.