Question

section 2.7: second derivative and concavity
score: 20/80 answered: 2/8
question 3
the function graphed above is:
concave up on the interval(s)
concave down on the interval(s)
there is an inflection point at:
question help: video

Explanation:

Step1: Recall concavity rules

If the second - derivative $f''(x)>0$, the function is concave up. If $f''(x)<0$, the function is concave down. Inflection points occur where $f''(x)$ changes sign.

Step2: Analyze the graph

Visually, a function is concave up when it "holds water" and concave down when it "spills water".
Looking at the graph, the function is concave up on the intervals $(-2,1)$ and $(3,\infty)$.
The function is concave down on the intervals $(-\infty,-2)$ and $(1,3)$.
Inflection points are where the concavity changes. The inflection points occur at $x = - 2,x = 1,x = 3$.

Answer:

Concave up on the interval(s): $(-2,1)\cup(3,\infty)$
Concave down on the interval(s): $(-\infty,-2)\cup(1,3)$
There is an inflection point at: $x=-2,x = 1,x = 3$