Question

given the graph of f(x) below, list where the inflection point of occurs and the intervals where f is concave up and concave down (enter your answers as a comma - separated lists of points or intervals, or enter dne for “does not exist” if there are none.) (you can click on a graph to enlarge it.) inflection points at: x = interval where function is concave up: interval where function is concave down:

Explanation:

Step1: Recall definitions

Inflection points are where concavity changes. Concave - up means the second - derivative $f''(x)>0$ and the graph curves upwards like a cup. Concave - down means $f''(x)<0$ and the graph curves downwards like an upside - down cup.

Step2: Identify inflection points

From the graph, we look for the points where the concavity changes. In this graph, the concavity changes at $x = 1$ and $x = 3$.

Step3: Determine concave - up interval

The function is concave up when the graph curves upwards. From the graph, the function is concave up on the intervals $(0,1)$ and $(3,5)$.

Step4: Determine concave - down interval

The function is concave down when the graph curves downwards. From the graph, the function is concave down on the interval $(1,3)$.

Answer:

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