Problem 40 - College Algebra with Corequisite Support

Question

For the following exercises, use a graphing utility to estimate the local extrema of each function and to estimate the intervals on which the function is increasing and decreasing. \(n(x)=x^{4}-8 x^{3}+18 x^{2}-6 x+2\)

Step-by-Step Solution

Answer

Use a graphing utility to find local extrema where the derivative is zero and identify intervals of increase/decrease by noting where the derivative is positive or negative.

Step 1: Find the derivative of the function

First, we need to find the derivative of the function since local extrema occur where the derivative is zero. Differentiate the function \( n(x) = x^{4} - 8x^{3} + 18x^{2} - 6x + 2 \) with respect to \( x \): \[ n'(x) = 4x^3 - 24x^2 + 36x - 6 \]

Step 2: Find critical points

Critical points occur where the derivative is zero or undefined. Solve for \( x \) in \( n'(x) = 0 \), resulting in:\[ 4x^3 - 24x^2 + 36x - 6 = 0 \] This can be solved using numerical methods or a graphing utility to find the approximate solutions for \( x \).

Step 3: Use graphing utility to find local extrema

Use a graphing utility to plot the derivative \( n'(x) \) and identify where it crosses the x-axis. These are the critical points. Estimate the x-values at these points to find the local extrema of \( n(x) \).

Step 4: Determine intervals of increase and decrease

Using the graph of \( n'(x) \), determine where it is positive (indicating \( n(x) \) is increasing) and where it is negative (indicating \( n(x) \) is decreasing). Note the intervals on the x-axis where these changes occur.

Key Concepts

Derivative

The derivative of a function is a fundamental concept in calculus that helps us understand how the function behaves. By finding the derivative, we can determine when the function is increasing or decreasing.

• The derivative represents the rate of change of the function's output with respect to its input.
• For a function like \( n(x) = x^{4} - 8x^{3} + 18x^{2} - 6x + 2 \), its derivative, denoted \( n'(x) \), is found using rules of differentiation.

In this case, applying these rules gives us the derivative \( n'(x) = 4x^3 - 24x^2 + 36x - 6 \). This derivative gives insights into the slope of the tangent line at any point on the curve of the function.Knowing the derivative allows us to proceed to find critical points, which ultimately helps in locating local extrema.

Critical Points

Critical points are essential for finding the local extrema, the highest and lowest points on a graph within a particular interval. They occur where the derivative of the function equals zero or is undefined.

• In our function's case, the critical points are solutions of the equation \( 4x^3 - 24x^2 + 36x - 6 = 0 \).
• Solving this polynomial equation reveals the x-values where the tangent is either flat (horizontal) or doesn't exist.

These points are crucial because they tell us potential turning points where the behavior of the function changes from increasing to decreasing, or vice versa. Using a graphing utility can simplify finding these values when solving these polynomial equations manually becomes too complicated.

Increasing and Decreasing Intervals

Determining where a function is increasing or decreasing is key to understanding its overall behavior. With the derivative in hand, we can analyze its sign across different intervals.

• The function is increasing where its derivative, \( n'(x) \), is positive.
• Conversely, it is decreasing where \( n'(x) \) is negative.

By examining the graph of the derivative \( n'(x) = 4x^3 - 24x^2 + 36x - 6 \), it's possible to determine specific x-intervals where these changes occur. This analysis tells us much about the function's behavior, particularly how it moves between rises and falls, which is useful for sketching its graph. Understanding these intervals helps in identifying patterns and trends depicted by the function.

Graphing Utility

A graphing utility is a powerful tool that aids in visualizing functions and performing complex calculations. When dealing with cumbersome polynomial equations like \( n(x) = x^{4} - 8x^{3} + 18x^{2} - 6x + 2 \), it becomes invaluable.

• It can plot both the original function and its derivative simultaneously, providing a comprehensive view of how the function behaves.
• By examining where the derivative crosses the x-axis, a graphing utility reveals the function's critical points.

These visuals make it easier to estimate where local extrema occur and distinguish increasing and decreasing intervals intuitively. In addition, interactive features in modern graphing utilities allow zooming and panning for precise readings, making it easier to comprehend complex curves. For students tackling calculus challenges, a graphing utility thus forms a bridge between abstract theoretical concepts and tangible understanding.