Question

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

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

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.

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 \(-6x + 6 = 0\).

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.