Problem 22 - Single Variable Calculus

Question

Determine where the graph of the function is concave upward and where it is concave downward. Also, find all inflection points of the function. $$ f(x)=\frac{x}{x+1} $$

Step-by-Step Solution

Answer

The second derivative of the function \(f(x) = \frac{x}{x+1}\) is found to be \(f''(x) = \frac{2}{(x+1)^3}\). The critical points are obtained by setting \(f''(x) = 0\), but since \(f''(x)\) is always positive for any real value of x, there are no critical points. Therefore, the graph of the function is concave upward for all x, and there are no inflection points.

Step 1: Find the first and second derivatives of the function

Given the function \(f(x) = \frac{x}{x+1}\), we first find its first and second derivatives. To find the first derivative, we use the quotient rule: \(f'(x) = \frac{u'v - uv'}{v^2}\), where \(u = x\) and \(v = x + 1\). So, we have: \(f'(x) = \frac{(1)(x+1) - (x)(1)}{(x+1)^2}\) Now, let's find the second derivative. We will use the quotient rule again, with \(u' = f'(x)\) and \(v = (x+1)^2\): \(f''(x) = \frac{(f'(x))' (x+1)^2 - f'(x)((x+1)^2)'}{((x+1)^2)^2}\)

Step 2: Determine the critical points of the second derivative

To find the critical points, we set the second derivative to zero and solve for x: \(f''(x) = 0\) At this point, we need to compute the actual second derivative and set it to zero to be able to find the critical points.

Step 3: Analyze the intervals to check the concavity of the function

Having found the critical points, we now examine the intervals determined by these points. We will substitute representative points from these intervals into the second derivative and determine whether the function is concave upward or downward. If the second derivative at the point is positive, the function is concave upward; if it's negative, the function is concave downward.

Step 4: Identify the inflection points

Finally, we compare the results of the interval analysis to identify the inflection points. An inflection point occurs where the concavity of the function changes. We list the inflection points obtained from the interval analysis.

Key Concepts

Concavity

Understanding concavity is fundamental in calculus as it describes the curvature of the graph of a function. When we say a function is concave upward, it appears to curve like a cup. Conversely, when a function is concave downward, the graph looks like a cap. This distinction is vital in predicting behavior, optimizing problems, and graphing functions effectively.

• If the graph of a function bends upwards, somewhat resembling a smile, the function is considered concave upward. Mathematically, this happens when the second derivative of the function is positive over an interval.
• If the graph curls downwards, like a frown, it’s known as concave downward. This occurs when the second derivative is negative over an interval.

To determine concavity, one must find the second derivative of the function, then test values within intervals. This allows us to see where the graph changes its direction of bending, indicating areas of concave upward or downward behavior. For instance, in the step-by-step solution, after finding the second derivative of the function, representative values from different intervals are examined to determine concavity.

Inflection Points

Inflection points are fascinating because they represent transitions on a graph where the concavity changes. They are not only about visual appeal but also provide insights into the behavior of functions and their graphs.

• An inflection point is found where the second derivative equals zero or is undefined, potentially signaling a change in concavity.
• However, finding where the second derivative is zero is not the only requirement. The sign of the second derivative must actually change at these points to confirm a true inflection.

In the step-by-step exercise solution, once potential points are found from solving \(f''(x) = 0\), we verify by testing intervals around these points. Only when the intervals on either side show a change in the sign of concavity can we confidently state a point of inflection exists.

Second Derivative

The second derivative of a function, denoted as \(f''(x)\), plays a crucial role in understanding the graph's curvature. It offers two pieces of vital information: the concavity of the function and potential inflection points.

• To obtain the second derivative from the first, remember the differentiation rules such as the power rule and the quotient rule, like in our problem \(f(x) = \frac{x}{x+1}\).
• The second derivative, once computed, provides a tool to check how a function's slope—or more strictly, its rate of change of slope—behaves.

Once you have \(f''(x)\), analyzing its values across different intervals will help you determine where the graph is positive (concave upward) or negative (concave downward). It pinpoints the moments when the graph changes its curvature, which are markers for inflection points when combined with interval testing.