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

Explanation:

Step1: Analyze function increasing

A function (y = f(x)) is increasing when its slope is positive. Looking at the graph, from (x = 0) to (x = 1), the function is decreasing. From (x=1) to (x = 3), the function is increasing. From (x = 3) to (x = 4), it is decreasing. From (x=4) to (x = 6), it is increasing. So the intervals of increase are ((0,1)) and ((4,6)).

Step2: Analyze function decreasing

A function (y = f(x)) is decreasing when its slope is negative. From (x = 1) to (x = 3), the function is increasing (not relevant for decreasing). From (x=3) to (x = 4), the function is decreasing. From (x = 0) to (x = 1), the function is decreasing. So the intervals of decrease are ((1,3)) and ((3,4)) is incorrect. The correct intervals of decrease are ((0,1)) and ((3,4)).

Step3: Analyze concave - upward

A function is concave - upward when the second derivative (f''(x)>0), which means the graph "holds water". Looking at the graph, from (x = 2) to (x = 6), the function is concave - upward. So the interval of concave - upward is ((2,6)).

Step4: Analyze concave - downward

A function is concave - downward when the second derivative (f''(x)<0), which means the graph "spills water". From (x = 0) to (x = 2), the function is concave - downward. So the interval of concave - downward is ((0,2)).

Answer:

(a) ((0,1)\cup(4,6)) (b) ((1,3)\cup(3,4)) (c) ((2,6)) (d) ((0,2))