Question

1. state the interval(s) on which the function shown below is concave up and concave down.

Explanation:

Step1: Recall concavity definitions

A function is concave up when its second - derivative is positive and concave down when its second - derivative is negative. Visually, a function is concave up when it "holds water" and concave down when it "spills water".

Step2: Analyze the graph

Looking at the graph, we can see that the function changes concavity at the points where the curve changes its curvature. The function is concave up on the intervals where the curve is shaped like a U - curve. The function is concave down on the intervals where the curve is shaped like an upside - down U - curve.
From the graph, the function is concave up on the intervals $(-\infty,- 1)$ and $(2,\infty)$. The function is concave down on the interval $(-1,2)$.

Answer:

Concave up: $(-\infty,-1)\cup(2,\infty)$; Concave down: $(-1,2)$