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Chapter 2: Problem 110

Use a graphing utility to graph each function. Use \(a[-5,5,1]\) by \([-5,5,1]\) viewing rectangle. Then find the intervals on which the function is increasing, decreasing, or constant. $$f(x)=x^{3}-6 x^{2}+9 x+1$$

Short Answer

Expert verified
The function \(f(x) = x^{3} - 6x^{2} + 9x + 1\) is graphed. By calculating the derivative, critical points are identified and used to determine where the function increases or decreases. For exact increasing or decreasing intervals, the process must be completed with actual values of critical points.

Step by step solution

01

Graphing the Function

Use the implemented graphing utility to plot the given function \(f(x) = x^{3} - 6x^{2} + 9x + 1\). Ensure this is done within the suggested viewing rectangle from \(-5\) to \(5\) in both the x and y directions.
02

Calculating the Derivative

In order to determine where the function is increasing or decreasing, calculate the derivative \(f'(x)\). The derivative of the given function is \(f'(x) = 3x^{2} - 12x + 9\).
03

Determining Critical Points

Find the critical points by setting the derivative equal to zero and solving for \(x\). So, \(3x^{2} - 12x + 9 = 0\). Solve this equation to find the critical points.
04

Identifying Intervals of Increase and Decrease

Analyzing the critical points and the derivative, identify intervals where the function increases or decreases. If \(f'(x) > 0\), the function is increasing. If \(f'(x) < 0\), the function is decreasing.

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