Question

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

Explanation:

Step1: Recall concavity and inflection - point concepts

The inflection - point is where the concavity of the function changes. A function is concave up when the second - derivative is positive and concave down when the second - derivative is negative.

Step2: Analyze the given information

We are given that the inflection point is at \(x = 1,3\). The function is concave up on the intervals \((0,1)\) and \((3,5)\). Then, by the nature of concavity and inflection points, the function must be concave down on the interval \((1,3)\).

Answer:

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