Question

1. try again

(you can click on a graph to enlarge it.)
inflection points at: $x = 1,3$
interval where function is concave up: $(0,1),(3,5)$
interval where function is concave down: $(1,3)$

Explanation:

Step1: Recall inflection - point definition

Inflection points are where the second - derivative changes sign. Visually, it's where the concavity changes.

Step2: Identify concavity

Concave up means the curve bends upward like a cup, and concave down means it bends downward like a cap.

Step3: Analyze the graph

By observing the graph, we can see that the concavity changes at approximately $x = 1$ and $x = 3$. So the inflection points are at $x=1,3$.

Step4: Determine concave - up intervals

The function is concave up on the intervals $(0,1)$ and $(3,5)$ as the curve bends upward on these intervals.

Step5: Determine concave - down interval

The function is concave down on the interval $(1,3)$ as the curve bends downward on this interval.

Answer:

Inflection points at: $x = 1,3$
Interval where function is concave up: $(0,1),(3,5)$
Interval where function is concave down: $(1,3)$