consider the following graph.\n(a) find the interval(s) on which ( f ) is increasing. (enter your answer…

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, we can see that the function is increasing on the intervals ((0.5,3)) 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, the function is decreasing on the intervals ((0,0.5)) and ((3,4)).

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 “holds water”. Looking at the graph, the function is concave upward on the interval ((2,5)).

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 “spills water”. From the graph, the function is concave downward on the intervals ((0,2)) and ((5,6)).

Answer:

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