Question

graphing the cubic function quick check
when given the function ( f(x)=x^{3} ), which of the following properties should the graph display? (1 point)
a graph that is only concave up.
a graph that is linear.
a graph that has a portion that is concave up and a portion that is concave down.
a graph that is only concave down.

Explanation:

Step1: Find the first and second derivatives

Given \(f(x)=x^{3}\).
The first derivative \(f^{\prime}(x) = 3x^{2}\) (using the power rule \((x^{n})^\prime=nx^{n - 1}\), here \(n = 3\)).
The second derivative \(f^{\prime\prime}(x)=6x\) (using the power rule on \(f^{\prime}(x)=3x^{2}\), with \(n = 2\)).

Step2: Analyze the concavity

Set \(f^{\prime\prime}(x)=0\), so \(6x = 0\) gives \(x = 0\).
When \(x<0\), say \(x=-1\), then \(f^{\prime\prime}(-1)=6\times(-1)=-6<0\), so the function is concave - down for \(x < 0\).
When \(x>0\), say \(x = 1\), then \(f^{\prime\prime}(1)=6\times1=6>0\), so the function is concave - up for \(x>0\).

A linear function has the form \(y = mx + b\) (degree 1), and \(f(x)=x^{3}\) is a cubic function (degree 3), so it is not linear.

Answer:

A graph that has a portion that is concave up and a portion that is concave down.