Problem 43 - Calculus of a Single Variable

Question

In Exercises \(43-46,\) use a computer algebra system to analyze the function over the given interval. (a) Find the first and second derivatives of the function. (b) Find any relative extrema and points of inflection. (c) Graph \(f, f^{\prime},\) and \(f^{\prime \prime}\) on the same set of coordinate axes and state the relationship between the behavior of \(f\) and the signs of \(f^{\prime}\) and \(f^{\prime \prime} .\) $$ f(x)=0.2 x^{2}(x-3)^{3}, \quad[-1,4] $$

Step-by-Step Solution

Answer

The relative extrema of the function \(f(x)=0.2 x^{2}(x-3)^{3}\) occur at \(x=0\) and \(x=3\), and the points of inflection occur at \(x=3/2, 3\). Observations from the graph of \(f\) and its derivatives show that when \(f^{\prime}\) is greater than zero, \(f\) is increasing; and when \(f^{\prime}\) is less than zero, \(f\) is decreasing. When \(f^{\prime\prime}\) is greater than zero, \(f\) is concave up; and when \(f^{\prime\prime}\) is less than zero, \(f\) is concave down.

Step 1: Find the first derivative of the function

By using the chain rule, and power rule, the first derivative of the function could be found as:\[f^{\prime}(x) = 0.2 [2x(x-3)^3 + x^2 *3(x-3)^{2}]\]

Step 2: Find the second derivative of the function

To find the second derivative, take the derivative of the first derivative:\[f^{\prime\prime}(x)= 0.2 [2(x-3)^3 + 12x(x-3)^{2} + 6x^2 (x-3)]\]

Step 3: Find any relative extrema and points of inflection

Set the first derivative to zero and solve for x to find points of relative extrema:\[0.2 [2x(x-3)^3 + x^2 *3(x-3)^{2}]=0\]The relative extrema occur at \(x=0\) and \(x=3\).Then, set the second derivative to zero and solve for x to find points of inflection:\[0.2 [2(x-3)^3 + 12x(x-3)^{2} + 6x^2 (x-3)]=0\]The points of inflection occur at \(x=3/2, 3\)

Step 4: Graph the function and its derivatives

Graph the function \(f(x)=0.2 x^{2}(x-3)^{3}\) and its first and second derivatives \(f^{\prime}(x)\) and \(f^{\prime\prime}(x)\) on the same set of coordinate axes. The signs of the first and second derivatives would tell us whether the function is increasing or decreasing, and whether it is concave up or concave down. From the graph, we can observe that when \(f^{\prime}\) is greater than zero, \(f\) is increasing; and when \(f^{\prime}\) is less than zero, \(f\) is decreasing. Similarly, when \(f^{\prime\prime}\) is greater than zero, \(f\) is concave up; and when \(f^{\prime\prime}\) is less than zero, \(f\) is concave down.

Key Concepts

First Derivative

The first derivative of a function, represented as \( f'(x) \), provides critical information about the rate at which the function's value is changing. Essentially, it tells us how the function’s output value (often representing position) will adjust in response to small changes in the input value (often time or distance).

In calculus, when you take the first derivative of a function, you're applying different rules like the power rule and the chain rule, as seen in our exercise, to differentiate polynomial functions such as \( f(x)=0.2x^2(x-3)^3 \). We find that \( f'(x) \) is positive when the function is increasing, and negative when it is decreasing. This transition is usually identified by setting \( f'(x) = 0 \), revealing potential relative maxima or minima, which are called relative extrema.

Second Derivative

The second derivative, \( f''(x) \), is essentially the derivative of the derivative, providing insight into the curvature or concavity of the original function. If the second derivative is positive at a certain point, the graph is concave up, resembling a U-shape there. Conversely, if it's negative, the graph is concave down, taking on an upside-down U shape.

Assessing the concavity is essential for determining points of inflection, which are points where the graph changes from being concave up to concave down, or vice versa. Finding \( f''(x) = 0 \) or where it's undefined, gives us a possible inflection point—as we've done in the solutions provided from our function \( f(x) \).

Relative Extrema

Relative extrema of a function refer to the topmost or bottommost points, locally, on the graph of the function. They are called 'relative' because they are the extreme values in their immediate vicinity as opposed to the absolute highest or lowest points on the entire graph. The first derivative test is a common method to find these points: if \( f'(x) \) changes from positive to negative, we have a relative maximum, and if \( f'(x) \) changes from negative to positive, there's a relative minimum.

In analyzing our exercise, we concluded that the relative extrema are located at the points where \( f'(x) \) is equal to zero—specifically at \( x=0 \) and \( x=3 \), with supporting calculation from the step-by-step solution.

Points of Inflection

Points of inflection are fascinating spots on a graph where the function's curvature changes direction. They are not only visually notable but also serve as a testament to the function's changing behavior. To search for these points, we move beyond the first derivative and into the realm of the second derivative, \( f''(x) \).

A change in sign of \( f''(x) \) signifies a change in concavity, and thus a potential inflection point. In the provided exercise solution, we determined the points of inflection by setting the second derivative equal to zero. In our specific case, \( x=3/2 \) and \( x=3 \) are points where the function changes concavity, offering insight into the underlying graph's structure.

Graphing Derivatives

Graphing derivatives is a visual representation of the concepts we discussed above. By sketching the curves of \( f(x) \), \( f'(x) \), and \( f''(x) \), we gain a deeper understanding of a function's behavior. Observing the graphs helps us associate the slopes of tangents and areas under the curve with the function’s increase, decrease, concavity, and convexity.

For example, as illustrated in the exercise solution, a positive slope on \( f'(x) \) indicates \( f(x) \) is rising, and a negative slope indicates it's falling. Furthermore, where \( f''(x) \) is positive, the graph of \( f(x) \) is curved upwards, and vice versa when negative. This integral relationship between a function and its derivatives is pivotal in analyzing motion, growth, and other dynamic changes.