Question

use the given graph of ( f ) over the interval ( (0,6) ) to find the following.
(a) the open intervals on which ( f ) is increasing. (enter your answer using interval notation.)
(b) the open intervals on which ( f ) is decreasing. (enter your answer using interval notation.)
(c) the open intervals on which ( f ) is concave upward. (enter your answer using interval notation.)
(d) the open intervals on which ( f ) is concave downward. (enter your answer using interval notation.)
(e) the coordinates of the point of inflection.
( (x,y)=(quad) )

Explanation:

Step1: Determine where \( f \) is increasing

A function \( y = f(x) \) is increasing when the slope of the tangent line is positive. Looking at the graph, \( f(x) \) is increasing on the intervals \((0.5,2)\) and \((4,6)\).

Step2: Determine where \( f \) is decreasing

A function \( y = f(x) \) is decreasing when the slope of the tangent line is negative. From the graph, \( f(x) \) is decreasing on the interval \((0,0.5)\) and \((2,4)\).

Step3: Determine where \( f \) is concave upward

A function \( y = f(x) \) is concave upward when the second - derivative \( f''(x)>0\), which means the graph of the function "holds water". From the graph, \( f(x) \) is concave upward on the interval \((3,6)\).

Step4: Determine where \( f \) is concave downward

A function \( y = f(x) \) is concave downward when the second - derivative \( f''(x)<0\), which means the graph of the function "spills water". From the graph, \( f(x) \) is concave downward on the interval \((0,3)\).

Step5: Determine the inflection point

An inflection point is a point where the concavity of the function changes. The concavity changes at \( x = 3\). To find the \( y \) - value, we look at the graph. When \( x = 3\), \( y=2\).

Answer:

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