Problem 23 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Find and classify all critical points. Determine whether or not \(f\) attains an absolute maximum and absolute minimum value. If it does, determine the absolute maximum and/or minimum value. \(f(x)=\frac{x-1}{x^{2}+3}\)

Step-by-Step Solution

Answer

The function has a critical point at \(x = -\frac{2}{3}\). The nature of this critical point (minimum, maximum, or neither) as well as whether the function attains an absolute maximum or minimum value depends on the behaviour of the function around this point, which is investigated in the final step.

Step 1: Calculating the derivative

We calculate the derivative of the function \(f(x) = \frac{x-1}{x^{2}+3}\) using the quotient rule. We have: \(f'(x) = \frac{(x^2+3) - (x-1)(2x)}{(x^2+3)^2}\) = \frac{3x+2}{(x^2+3)^2}\

Step 2: Finding critical points

Next, set the derivative equal to zero to find the critical points. So: \(\frac{3x+2}{(x^2+3)^2} = 0\). Therefore, our critical points are \(x = -\frac{2}{3}\).

Step 3: Checking for maximum and minimum values

To know whether these points are maximum or minimum, we have to find the second derivative and substitute our critical points into the second derivative. If the result is less than 0, then we have a local maximum. If it is greater than 0, then we have a local minimum. Also, to check whether the function attains an absolute maximum or minimum value, we have to plug in the endpoints (if any) and critical points into the original function, and compare those values.

Key Concepts

Quotient Rule

The quotient rule is an essential tool in calculus, used when you're taking the derivative of a fraction where both the numerator and the denominator are functions of a variable, typically denoted as \(u(x)\) and \(v(x)\), respectively. In essence, to find the derivative of \( \frac{u}{v} \) where both \(u\) and \(v\) are functions of \(x\), the quotient rule states:

\[ f'(x) = \frac{u'v - uv'}{v^2} \
\] where \(u'\) and \(v'\) are derivatives of \(u\) and \(v\), respectively. For the function \(f(x)=\frac{x-1}{x^{2}+3}\), applying the quotient rule involved differentiating both the numerator (\(x-1\)) and the denominator (\(x^{2}+3\)) independently. Remember, when using the quotient rule, it's critical to ensure proper calculation of both derivatives and careful subtraction and factorial expansion to simplify the resultant derivative correctly. Missteps in any of these places can lead to incorrect critical points, which in turn affect the determination of maximum and minimum values.

Derivative Calculation

Derivative calculation is a fundamental operation in calculus that measures how a function changes as its input changes. It's like examining the slope of a curve at any given point. The derivative provides us with a mathematical way of finding rates of change and is used to find critical points, which are candidates for where a function might have a maximum or minimum value. To differentiate a function means to calculate its derivative.

For the function in question, \(f(x)\), we used the quotient rule to get the derivative function \(f'(x)\). This step was critical since the next process of finding critical points hinges on the information provided by the first derivative. Derivative calculation can involve straightforward rules such as the power rule or more complex rules such as the product or chain rules, depending on the form of the original function.

Maximum and Minimum Values

Finding the maximum and minimum values of a function is a principal application of derivatives in calculus. Critical points are locations on the graph of a function where its derivative \(f'(x)\) is zero or undefined, and they are potential candidates for local (or global) maxima or minima.

After calculating the derivative and finding the critical points, we use the second derivative test to determine whether each critical point corresponds to a local maximum or minimum. This involves taking the derivative of the derivative \(f''(x)\), known as the second derivative. If \(f''(x) > 0\) at a critical point, the function has a local minimum there; if \(f''(x) < 0\), there's a local maximum. If \(f''(x) = 0\), the test is inconclusive.

Lastly, to determine the absolute maximum and minimum on a closed interval, we also consider the function's values at endpoint(s), if they exist. Since the function in our example is not provided with an interval, we would assume it extends over all real numbers and look at the behavior of \(f(x)\) as \(x\) approaches positive or negative infinity to discuss absolute maximums and minimums. This thorough investigation ensures we fully understand the behavior and characteristics of the function across its entire domain.