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Chapter 9: Problem 12

Sketch the graph of the function. Choose a scale that allows all relative extrema and points of inflection to be identified on the graph. \(y=-x^{3}+3 x^{2}+9 x-2\)

Short Answer

Expert verified
The graph of the function \(y=-x^{3}+3 x^{2}+9 x-2\) is a curve that shows all relative extrema and inflection points. We identify these using the first derivative \(y'= -3x^{2} + 6x + 9\) and second derivative \(y''= -6x + 6\).

Step by step solution

01

First Derivative and Critical Points

Differentiate the function \(y=-x^{3}+3 x^{2}+9 x-2\) to find the first derivative, \(y'= -3x^{2} + 6x + 9\). The critical points can be found by setting the derivative equal to zero and solve for \(x\). The solutions are the x-coordinates of the potential maxima and minima.
02

Second Derivative and Inflection Points

Differentiate the first derivative to get the second derivative: using \(y' = -3x^{2} + 6x + 9\), we find \(y''= -6x + 6\). Points of inflection are given by the solutions of \(y'' = 0\), thus, we need to solve \(-6x + 6 = 0\).
03

Draw the Function Graph

Using the critical points and the points of inflection noted, take additional points to help in plotting the function to get a complete picture. Start by identifying the x and y-intercepts. Then plot the graph, making sure to show all extrema and inflection points.

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Most popular questions from this chapter

Sketch the graph of the function. Choose a scale that allows all relative extrema and points of inflection to be identified on the graph. \(y=-4 x^{3}+6 x^{2}\)

Compare the values of \(d y\) and \(\Delta y\). \(y=2 x+1 \quad x=2 \quad \Delta x=d x=0.01\)

Sketch the graph of the function. Choose a scale that allows all relative extrema and points of inflection to be identified on the graph. \(y=x^{4}-4 x^{3}+16 x\)

Let \(x=2\) and complete the table for the function. $$ \begin{array}{|c|c|c|c|c|} \hline d x=\Delta x & d y & \Delta y & \Delta y-d y & d y / \Delta y \\ \hline 1.000 & & & & \\ \hline 0.500 & & & & \\ \hline 0.100 & & & & \\ \hline 0.010 & & & & \\ \hline 0.001 & & & & \\ \hline \end{array} $$ \(y=\frac{1}{x^{2}}\)

Use a graphing utility to graph the function. Choose a window that allows all relative extrema and points of inflection to be identified on the graph. \(y=x^{1 / 3}+1\)
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