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. \(y=2-x-x^{3}\)

Short answer

The graph of \(y = 2 - x - x^{3}\) will intersect the x-axis at only one point. It will have critical points where the derivative equals zero or is undefined. The inflection points will be where the second derivative equals zero. The graph's end behavior will show that as \(x\) goes to positive or negative infinity, \(y\) goes to negative infinity.

Step-by-step solution

Find the Intercepts

Set \(y=0\), then solve for \(x\). This equation becomes \(0=2-x-x^{3}\). Solving it will give the x-intercept(s). Similarly, put \(x=0\) to obtain the y-intercept.

Determine Critical and Inflection Points

Derive once to get \(y'= -1 - 3x^{2}\). This derivative gives the slope of the tangent line at any point of the function. For critical points, we find the \(x\) at which \(y'\) is either 0 or undefined. Solve \(-1 - 3x^{2} = 0\). Next, derive twice to get \(y'' = -6x\). Then find the x-value where \(y'' = 0\). These are the inflection points.

Apply End Behavior

As \(x\) goes to positive or negative infinity, we need to understand what happens to \(y\). This will tell us how the function behaves at its bounds.

Sketch the Graph

Plot the intercepts, critical points, inflection points, and apply the end behavior to create a complete sketch of the graph. Ensuring that you've identified all relative extrema and points of inflection.