Problem 26 - Essential Calculus

Question

Find the critical numbers of \(f\) (if any). Find the open intervals on which the function is increasing or decreasing and locate all relative extrema. Use a graphing utility to confirm your results. $$ f(x)=x^{4}-32 x+4 $$

Step-by-Step Solution

Answer

The critical number of the function \(f(x) = x^{4} - 32x + 4\) is \(x = 2\). The function is decreasing on the interval \((-\infty, 2)\) and increasing on the interval \((2, \infty)\). The relative minimum is at \(x = 2\).

Step 1: Find the derivative of the function

The derivative of \(f(x) = x^{4} - 32x + 4\) is \(f'(x) = 4x^{3} - 32\).

Step 2: Find the critical numbers

Set the derivative equal to zero to find the critical numbers: \(4x^{3} - 32 = 0\). Solving for x, we get \(x = \sqrt[3]{8}\), which is about \(x = 2\). The critical numbers of f are the x values for which f'(x) = 0 and f''(x) exists, so the critical number is \(x = 2\).

Step 3: Determine the intervals of increase and decrease

Plot the critical number on a number line and use test points to determine where \(f'\) is positive (indicating increase) or negative (indicating decrease). For \(x < 2 \), try \(x = 0\). \(f'(0) = -32 < 0\), so \(f\) is decreasing on \((-\infty, 2)\). For \(x > 2\), try \(x = 3\). \(f'(3) = 68 > 0\), so \(f\) is increasing on \((2, \infty)\).

Step 4: Find the relative extrema

The relative extremum is where the function changes from increasing to decreasing or vice versa. Here, the function changes from decreasing to increasing at \(x = 2\). So, \(f\) has a relative minimum at \(x = 2\).

Step 5: Confirm with a graphing utility

A graphing utility can be used to graph the function \(f(x) = x^{4} - 32x + 4\), and it clearly shows the relative minimum at \(x = 2\).

Key Concepts

Intervals of Increase and Decrease

When working with a function, one of the main tasks is to identify where it is increasing or decreasing. This helps you understand how the function behaves over certain ranges. To find these intervals, we analyze the sign of the derivative of the function.

• First, find the derivative of the function, denoted as \( f'(x) \). In this case, the derivative is \( f'(x) = 4x^3 - 32 \).
• The critical numbers where the function may change behavior are found by setting the derivative to zero \( 4x^3 - 32 = 0 \). This gives us a critical number of \( x = 2 \).
• Use the critical number to divide the number line into intervals: \((-\infty, 2)\) and \((2, \infty)\).
• Select test points within each interval to substitute back into the derivative.
  • If \( f'(x) \) is positive, the function is increasing in that interval.
  • If \( f'(x) \) is negative, the function is decreasing in that interval.

By checking these intervals, we established that \( f(x) \) is decreasing on \((-\infty, 2)\) and increasing on \((2, \infty)\). This provides valuable insight into the shape of the graph and predicts how the function behaves relative to changes in the x-values.

Relative Extrema

Relative extrema are the points on the graph where the function has a relative minimum or maximum. These are places where the function changes direction from increasing to decreasing or decreasing to increasing.

• A relative minimum occurs where the function changes from decreasing to increasing. This corresponds to the trough of the graph.
• A relative maximum occurs where the function changes from increasing to decreasing, corresponding to the peak of the graph.

For the function \( f(x) = x^4 - 32x + 4 \), the relative extremum is at the critical number \( x = 2 \). At this point, the function changes from decreasing on \((-\infty, 2)\) to increasing on \((2, \infty)\).This indicates that there's a relative minimum at \( x = 2 \). By analyzing this, you are equipped to estimate where the high and low points occur on a graph, which helps when sketching or analyzing graphs.

Graphing Utilities

A graphing utility is a tool that allows you to visualize the graph of a function, verifying your analytic results by providing a graphical perspective.

• Graphing utilities, like graphing calculators or computer software, can plot the function for you.
• These tools confirm the calculated critical points and intervals of increase and decrease by showing the actual behavior of the function graphically.

For example, using a graphing utility to plot \( f(x) = x^4 - 32x + 4 \), you would clearly see a minimum point at \( x = 2 \), where the graph dips before rising again. This visualization confirms the analytical solution and allows users to better understand complex functions at a glance.Having a graphical representation is beneficial, as it provides visual confirmation of where the function increases or decreases, and it reinforces the understanding of relative extrema through live data visualization.