(a) find the interval(s) on which ( f ) is increasing. (enter your answer using interval notation.)\n(b)…

Explanation:

Step1: Determine where the function is increasing

A function (y = f(x)) is increasing when its slope (derivative) is positive. Looking at the graph, we can see that the function is increasing on the intervals ((0.5,3)) and ((4,6)).

Step2: Determine where the function is decreasing

A function (y = f(x)) is decreasing when its slope (derivative) is negative. From the graph, the function is decreasing on the intervals ((0,0.5)) and ((3,4)).

Step3: Determine where the function is concave upward

A function (y = f(x)) is concave upward when the second - derivative is positive. Visually, this is where the graph “holds water”. The function is concave upward on the interval ((2,5)).

Step4: Determine where the function is concave downward

A function (y = f(x)) is concave downward when the second - derivative is negative. Visually, this is where the graph “spills water”. The function is concave downward on the intervals ((0,2)) and ((5,6)).

Step5: Find the points of inflection

Points of inflection occur where the concavity changes. The concavity changes at (x = 2) and (x=5). When (x = 2), (y = 3) and when (x = 5), (y = 4). So the points of inflection are ((2,3)) and ((5,4)).

Answer:

(a) ((0.5,3)\cup(4,6)) (b) ((0,0.5)\cup(3,4)) (c) ((2,5)) (d) ((0,2)\cup(5,6)) (e) ((2,3),(5,4))