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

Short answer

Due to the complexity of solving the cubic function, the roots obtained might be unrealistic to solve without the aid of a calculator or software. After finding the roots, determine the first and second derivative which helps in finding local maxima, local minima, and inflection points. The concavity and intervals of increase and decrease are then verified to develop an accurate sketch.

Step-by-step solution

Find the Roots

Solve \(y=-x^{3}+x-2=0\) for \(x\). Here we have to find the values of \(x\) that make the \(y\) value 0. Unfortunately, cubic equations cannot be solved easily by factoring or using the quadratic formula. So, these roots may be found by graphical methods, numerical methods or by using the cubic formula if you are familiar with it.

Find the Derivatives

Find the first and second derivatives of the function. The first derivative shows where the function has local extrema (maximum or minimum), and the second derivative shows where the function has inflection points. \nThe first derivative \(y' = -3x^{2}+1\). Set this to zero to find critical points: \(-3x^{2}+1=0\). Solve this for \(x\), which will give you the x-coordinates of the extrema. \nThe second derivative is \(y'' = -6x\). Set this to zero to find possible inflection points: \(-6x=0\). Solve this for \(x\), which will give you the x-coordinate of inflection points.

Determine the Behavior at Extrema and Inflection Points

Plug each critical point and potential inflection point into the original function to find their \(y\) values. This will give you the points for the extrema and points of inflection. Also look at the function between each of these x-values (using a number line can be useful here), to determine the behavior of the function.

Sketch the Graph

Finally, using all the information gathered, sketch the function. Be sure to accurately plot the roots, extrema, and points of inflection. The function's behavior between these points (increasing/decreasing, concave up/concave down) will help guide the overall shape of the sketch.