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. \(y=x^{5}+1\)

Short answer

The function \(y=x^{5}+ 1\) has no relative extrema and no points of inflection. The graph is always increasing and it looks like curve shifted one unit upwards from typical \(x^{5}\).

Step-by-step solution

Compute the First Derivative

The first derivative of the function \(y=x^{5}+1\) is found using the Power Rule of Derivatives. Hence the derivative \(y'\) can be calculated as \(y'=5x^{4}\)

Find the Critical Points

Set the first derivative equal to zero to find the critical points: \(5x^{4}=0\). The only solution to this equation is \(x=0\)

Compute the Second Derivative

Second derivative is derivative of first derivative which is: \(y''=20x^{3}\)

Identify the Inflection Points

To find inflection points, \(y''=20x^{3}=0\). The only solution to this equation is \(x=0\). However, because the concavity does not change sign, there are no points of inflection

Sketch the Graph of the Function

There are no relative extrema and no points of inflection, and the function is always increasing. You thus expect the graph to look like a curve that begins below the x-axis and is always increasing. It is shifted one unit above from typical \(x^{5}\) curve due to additional \(+1\) in original function