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 po
a graph that has a portion that is concave up and a portion that is concave down.
a graph that is linear.
a graph that is only concave down.
a graph that is only concave up.

Explanation:

Step1: Find the second derivative

To determine concavity, we find the second derivative of \( f(x) = x^3 \). First derivative: \( f'(x) = 3x^2 \). Second derivative: \( f''(x) = 6x \).

Step2: Analyze concavity

• When \( x > 0 \), \( f''(x) = 6x > 0 \), so the function is concave up.
• When \( x < 0 \), \( f''(x) = 6x < 0 \), so the function is concave down.

Thus, the graph of \( f(x) = x^3 \) has a portion concave up and a portion concave down.

Answer:

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