Question

given the graph of f(x) below, list where the inflection point of f 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 is positive and concave - down means the second - derivative is negative.

Step2: Analyze the graph

Visually inspect the graph for where the curve changes from curving upward to curving downward or vice - versa.

Step3: Determine inflection points

The inflection points occur where the slope of the tangent line to the curve of the first - derivative has a maximum or minimum. From the graph, assume the inflection point is at $x = a$ (a value that can be read from the graph).

Step4: Find concave - up intervals

The function is concave up where the graph looks like a "cup" opening upward. Let the interval be $(b,c)$.

Step5: Find concave - down intervals

The function is concave down where the graph looks like a "cap" opening downward. Let the interval be $(d,e)$.

Answer:

Inflection points at: $x =$ <value from graph>, Interval where function is concave up: <concave - up interval>, Interval where function is concave down: <concave - down interval>

(Note: Without the actual values from the graph, we can't give specific numerical answers. But the process to find them is as above. If there are no inflection points, write DNE for the inflection - point part, and similar for the concave - up and concave - down intervals if they don't exist.)