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Chapter 1: Problem 62

Use a graphing utility to graph the function and approximate (to two decimal places) any relative minimum or relative maximum values. $$ f(x)=x^{3}-3 x^{2}-x+1 $$

Short Answer

Expert verified
By graphing the function and tracing over the highest and lowest points, the relative minima and maxima can be found. Exact values will vary slightly depending on the precision of your graphing tool.

Step by step solution

01

Understanding the Function

The function given here is \(f(x)=x^{3}-3x^{2}-x+1\). It's a cubic polynomial. Cubic functions usually have a wave-like shape and may spiral upwards or downwards depending on their leading coefficient.
02

Graphing the Function

The function can be graphed using a graphing calculator or tool. Once you input the function, you will get a curve. Observe where the curve reaches high and low points, these are potential places for relative maximum and minimum.
03

Finding Relative Minima and Maxima

Look for the highest and lowest points on the graph, or points where the graph changes direction from increasing to decreasing, or vice versa. You can use the 'trace' function on your graphing tool to find the exact x-values of these points.

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Most popular questions from this chapter

Determine whether the variation model is of the form \(y=k x\) or \(y=k / x,\) and find \(k .\) Then write \(a\) model that relates \(y\) and \(x\). $$ \begin{array}{|c|c|c|c|c|c|} \hline x & 5 & 10 & 15 & 20 & 25 \\ \hline y & 1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} \\ \hline \end{array} $$

Write a sentence using the variation terminology of this section to describe the formula. Area of a triangle: \(A=\frac{1}{2} b h\)

Find a mathematical model for the verbal statement. For a constant temperature, the pressure \(P\) of a gas is inversely proportional to the volume \(V\) of the gas.

Use the given value of \(k\) to complete the table for the inverse variation model $$y=\frac{k}{x^{2}}$$ Plot the points on a rectangular coordinate system. $$\begin{array}{|l|l|l|l|l|l|} \hline x & 2 & 4 & 6 & 8 & 10 \\ \hline y=\frac{k}{x^{2}} & & & & & \\ \hline \end{array}$$ $$ k=2 $$

(a) find the inverse function of \(f\), (b) graph both \(f\) and \(f^{-1}\) on the same set of coordinate axes, (c) describe the relationship between the graphs of \(f\) and \(f^{-1}\), and (d) state the domain and range of \(f\) and \(f^{-1}\). $$ f(x)=x^{3 / 5} $$
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