Question

In Exercises, you are given \(f^{\prime}\). Find the intervals on which (a) \(f^{\prime}(x)\) is increasing or decreasing and (b) the graph of \(f\) is concave upward or concave downward. (c) Find the relative extrema and inflection points of \(f\). (d) Then sketch a graph of \(f\). $$ f^{\prime}(x)=x^{2}+x-6 $$

Short answer

The function \(f^{\prime}(x)=x^{2}+x-6\) increases on the interval \((-3,2)\) and decreases on the intervals \((-\infty,-3)\) and \((2,\infty)\). Its second derivative \(f^{\prime\prime}(x)=2x+1\) indicates that the function is concave upward on the interval \((-\frac{1}{2},\infty)\) and concave downward on the interval \((-\infty, -\frac{1}{2})\). It has a relative maximum at \(x=-3\), a relative minimum at \(x=2\), and an inflection point at \(x=-\frac{1}{2}\). The graph of \(f(x)\) can be sketched using these points and intervals. The function \(f(x)\) can be found by integrating \(f^{\prime}(x)\), and it's \(\frac{1}{3}x^{3}+\frac{1}{2}x^{2}-6x+C\).

Step-by-step solution

Find the intervals where \(f^{\prime}(x)\) is increasing or decreasing

Determine the zeros of \(f^{\prime}(x)\) by solving the equation \(f^{\prime}(x)=0\). This gives \(x^{2}+x-6=0\), which factors into \((x-2)(x+3)=0\). Hence the solutions are \(x=2\) and \(x=-3\). Therefore, \(f^{\prime}(x)\) increases on the interval \((-3,2)\) and decreases on the intervals \((-\infty,-3)\) and \((2,\infty)\).

Determine concavity and points of inflection

Find the second derivative \(f^{\prime\prime}(x)\) by differentiating \(f^{\prime}(x)\) which gives \(f^{\prime\prime}(x)=2x+1\). Setting it to zero determines the points of inflection: \(2x+1=0\), hence \(x=-\frac{1}{2}\). The function is concave upward when \(f^{\prime\prime}(x)>0\) which is on the intervals \((-\frac{1}{2}, \infty)\) and concave downward when \(f^{\prime\prime}(x)<0\) which is on the interval \((-\infty, -\frac{1}{2})\)

Find relative extrema and inflection point

The local minimum and maximum occur at the critical numbers where the sign of the first derivative changes. From Step 1, we know these are at \(x=-3\) and \(x=2\), which are the relative extrema dependent on the original function's values at these points. The inflection point is at \(x=-\frac{1}{2}\) since it is where the sign of \(f^{\prime\prime}(x)\) changes.

Sketch the graph

Use the points and intervals from the previous steps. Start by plotting the points \((-3,f(-3))\), \((2,f(2))\) and \((-\frac{1}{2}, f(-\frac{1}{2}))\). Draw the graph concave up on \((-\frac{1}{2}, \infty)\) and concave down on \((-\infty,-\frac{1}{2})\) noticing that there are maximum and minimum at \(x=-3\) and \(x=2\) respectively.

Final Integration

Finally, integrate \(f^{\prime}(x) = x^{2}+x-6\) to find \(f(x)\). This gives \(f(x)=\frac{1}{3}x^{3}+\frac{1}{2}x^{2}-6x+C\) (where \(C\) is the integration constant). Determine \(C\) using an arbitrary point on the graph or it could be left unspecified if it's unneeded.

Key concepts

First Derivative Test

The First Derivative Test is invaluable for identifying intervals where a function is increasing or decreasing. To perform this test, you first need to find the critical points where the derivative is equal to zero or undefined. In this case, the function we are considering is the derivative itself, so we set \(f^{\prime}(x)=x^{2}+x-6\) equal to zero.
The solutions, \(x=2\) and \(x=-3\), are key in determining where the function changes from increasing to decreasing or vice versa. By analyzing the sign of the derivative in intervals around these critical points—such as entering values into the derivative expression—you can deduce:

• Increasing on \((-3, 2)\)
• Decreasing on \((-\infty, -3) \) and \( (2, \infty)\).

This analysis is crucial to spotting the relative maxima or minima, confirming changes in direction of a function at given points.

Concavity

Concavity helps us understand how the shape of the graph behaves, indicating whether a curve bends upward or downward. To determine the concavity of the function, we use the second derivative, which reflects how the slope of the tangent line changes.
For our function, the second derivative is \(f^{\prime\prime}(x)=2x+1\). By setting this equal to zero, we find the potential inflection point at \(x=-\frac{1}{2}\).

• The function is concave up (like a smile) on the interval \((-\frac{1}{2}, \infty)\), where \(f''(x)>0\).
• Conversely, it is concave down (like a frown) on \((-\infty, -\frac{1}{2})\), where \(f''(x)<0\).

Concavity plays a significant role in the behavior of the graph, especially around inflection points, where the concavity changes.

Inflection Points

Inflection points are where a graph changes its concavity. To find these points, determine where the second derivative equals zero and examine the sign changes around these points. For our function, the second derivative is \(2x+1\) and the inflection point occurs at \(x = -\frac{1}{2}\) where the second derivative changes sign.
At this point, the graph transitions from being concave downward to concave upward. It signifies a shift in the tower or the concave structure of the graph. Thus, inflection points are important for understanding the flexibility and shape dynamics of curves in graphical analysis.

Graph Sketching

Graph sketching is the ultimate step where all previous analyses come together. With the critical points \(x=-3\) and \(x=2\), as well as the inflection point \(x=-\frac{1}{2}\), you can build a clear picture of what the graph looks like.
Start by plotting these key points on a graph:

• Observe concavity changes by noting that between \(x=-\frac{1}{2}\) and moving towards \infty, the graph is concave upward.
• Similarly, before \(x=-\frac{1}{2}\), it is concave downward, allowing for a smooth transition at the inflection point.
• Consider the maxima and minima revealed by the gradient changes, ensuring to connect these points with smooth curves.

Lastly, remember that integrating the derivative allows you to express the original function, showing its path clearly through calculated positions. With these plotted concavity, extrema, and direction details, you can effectively sketch the curve, displaying all its subtleties and transitions.