Problem 1 - Thomas Calculus

Question

Answer the following questions about the functions whose derivatives are given: \begin{equation}\begin{array}{l}{\text { a. What are the critical points of } f ?} \\ {\text { b. On what open intervals is } f \text { increasing or decreasing? }} \\ {\text { c. At what points, if any, does } f \text { assume local maximum and }} \\ \quad {\text { minimum values? }}\end{array}\end{equation} \begin{equation}f^{\prime}(x)=x(x-1)\end{equation}

Step-by-Step Solution

Answer

Critical points are at \(x=0\) and \(x=1\). \(f\) increases on \((-\infty, 0)\) and \((1, \infty)\), and decreases on \((0, 1)\). Local maximum at \(x=0\) and minimum at \(x=1\).

Step 1: Find Critical Points

To find the critical points of the function, set the derivative \(f'(x) = x(x-1)\) equal to zero. Solve the equation \(x(x-1) = 0\), which gives the critical points as \(x = 0\) and \(x = 1\).

Step 2: Determine Intervals

Use the critical points to determine intervals by setting x-values around the critical points. The intervals will be \((-\infty, 0)\), \((0, 1)\), and \((1, \infty)\).

Step 3: Test Increasing or Decreasing Nature

For each interval, select a test point not equal to the critical points and substitute it into the derivative \(f'(x) = x(x-1)\) to determine sign. - For \((-\infty, 0)\), choose \(x = -1\), and \(f'(-1) = (-1)((-1)-1) = 2\), so it is increasing.- For \((0, 1)\), choose \(x = \frac{1}{2}\), and \(f'(\frac{1}{2}) = (\frac{1}{2})(\frac{1}{2}-1) = -\frac{1}{4}\), so it is decreasing.- For \((1, \infty)\), choose \(x = 2\), and \(f'(2) = 2(2-1) = 2\), so it is increasing.

Step 4: Identify Local Extrema

Use the first derivative test by analyzing the change of sign of \(f'(x)\) around the critical points.- At \(x = 0\), \(f'(x)\) changes from positive to negative, so \(x = 0\) is a local maximum.- At \(x = 1\), \(f'(x)\) changes from negative to positive, so \(x = 1\) is a local minimum.

Key Concepts

Derivatives

In calculus, a derivative represents the rate at which a function changes with respect to one of its variables. Essentially, the derivative of a function at a point gives you the slope of the tangent line to the function's graph at that point. This concept is crucial for understanding many behaviors of functions.

• The derivative of a function can provide information on the function's growth and rate of change.
• For a given function \( f(x) \), the derivative \( f'(x) \) is generally found using differentiation rules like the power rule, product rule, quotient rule, and chain rule.
• The critical points of a function, where the derivative equals zero or is undefined, are key in analyzing the function's behavior.

Understanding derivatives is the first step in exploring more complex properties of functions like their shape and turning points.

Increasing and Decreasing Intervals

Identifying where a function is increasing or decreasing is fundamental for understanding the overall trend of the function. This behavior is determined by examining the sign of the derivative.

• A function is increasing on an interval if its derivative is positive across that interval.
• Conversely, a function is decreasing on an interval if the derivative is negative across that interval.
• Critical points often serve as boundary markers that separate increasing and decreasing behaviors.

To properly assess a function’s intervals, the derivative is tested at points between critical points. For example, choosing a test point in the interval \( (0, 1) \) for the derivative \( f'(x) = x(x-1) \) helps determine if the function is increasing or decreasing on that interval.

Local Extrema

Local extrema of a function, which include local minimums and maximums, are points where the function values are the least or greatest, respectively, in a small neighborhood. Discovering these points is essential as they help in sketching the graph of a function.

• A local maximum occurs at a critical point if the function changes from increasing to decreasing.
• A local minimum happens where the function shifts from decreasing to increasing.
• Analyzing the sign change of the derivative around critical points helps identify these extrema.

In our problem, the critical point \( x = 0 \) is a local maximum, since the derivative changes from positive to negative here. Similarly, \( x = 1 \) is a local minimum due to the derivative transitioning from negative to positive.

First Derivative Test

The first derivative test is a straightforward method used to classify the local extrema of a function. This test uses the derivative's sign change over critical points to determine where local minimums and maximums occur.

• If the derivative changes from positive to negative at a critical point, the point is a local maximum.
• If the derivative switches from negative to positive, the point is a local minimum.
• No change in sign might indicate a point of inflection rather than an extrema.

By applying the first derivative test to \( f'(x) = x(x-1) \), one can clearly see where the local extrema lie, aiding in understanding the function's graph more fully. Observing the test results in change patterns provides insights into the function's behavior near its critical points.