Question

Use a graph to estimate the critical numbers of \(f\left( x \right) = \left| {{\bf{1}} + {\bf{5}}x - {x^3}} \right|\) correct to one decimal place.

Short answer

There are five critical numbers \(x = \pm 1.3,{\rm{ }} - 2.1,{\rm{ }} - 0.2,{\rm{ and }}2.3\) for \(f\left( x \right)\).

Step-by-step solution

Sketch the graph of \(f\left( x \right)\)

In the desmos graphing calculator enter the expression \(\left| {1 + 5x - {x^3}} \right|\) in the tab to plot the curve of \(f\left( x \right) = \left| {1 + 5x - {x^3}} \right|\).

The figure below represents the graph of \(f\left( x \right)\).

Find the critical numbers of \(f\left( x \right)\)

Check the numbers where \(f'\left( x \right)\) is not defined. From the graph it can be observed that \(f'\left( x \right)\) does not exist at \(x \approx - 2.1,{\rm{ }} - 0.2,{\rm{ and }}2.3\).

From the graph the solution of the equation \(f'\left( x \right) = 0\) are \(x = \pm 1.3\).

Thus, there are five critical numbers for \(f\left( x \right)\) which are \(x = \pm 1.3,{\rm{ }} - 2.1,{\rm{ }} - 0.2,{\rm{ and }}2.3\).