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. \(y=x^{5}-5 x\)

Short answer

The graph of the function \(y=x^{5}-5x\) can be sketched by plotting the intercepts, extrema points, and inflection points at \(x = 0\) and \(x = \pm \sqrt[4]{5}\). The graph increases without bound (to positive and negative infinity) as \(x\) moves away from 0.

Step-by-step solution

Find the x- and y-intercepts

Find the x-intercepts by setting \(y=0\) and solving for \(x\), the result being \(x = 0\) and \(x = \pm \sqrt[4]{5}\). The y-intercept is found by setting \(x=0\) and solving for \(y\) resulting in \(y=0\).

Find the extrema points

The extrema points are found by taking the derivative of the function and setting it equal to zero, \(0 = 5x^{4}-5 = 0\), and solving for \(x\). This gives the same values as the x-intercepts, which can be confirmed by checking that the second derivative at these points is non-zero. Hence, \(x = 0\) and \(x = \pm \sqrt[4]{5}\) are local extrema points.

Find the inflection points

The points of inflection can be found by taking the second derivative and setting it equal to zero. The second derivative of the function is \(20x^{3}\), and setting this equal to zero results in \(x = 0\).

Check the behavior towards infinity

As \(x\) approaches positive or negative infinity, so does \(y\). This means the major axis of the graph is the y-axis.

Sketch the graph

Plot the x- and y-intercepts, extrema points, and inflection points. Connect these points smoothly knowing the graph increases without bound (to positive and negative infinity) as \(x\) moves away from 0.