Question

(a) find the interval(s) on which ( f ) is increasing. (enter your answer using interval notation.)
(b) find the interval(s) on which ( f ) is decreasing. (enter your answer using interval notation.)
(c) find the open interval(s) on which ( f ) is concave upward. (enter your answer using interval notation.)
(d) find the interval(s) on which ( f ) is concave downward. (enter your answer using interval notation.)
(e) find the coordinates of the point(s) of inflection.
( (x, y)=(quad) )

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) \((2,3),(5,4)\)