Question

consider the following graph. (a) find the interval(s) on which ( f ) is increasing. (enter your answer using interval notation.) ( (1,3),(4, infty) ) (b) find the interval(s) on which ( f ) is decreasing. (enter your answer using interval notation.) ( (-infty, 1),(3,4) ) (c) find the open interval(s) on which ( f ) is concave upward. (enter your answer using interval notation.) ( (2, infty) ) (d) find the interval(s) on which ( f ) is concave downward. (enter your answer using interval notation.) ( (-infty, 2) )

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) \((0,2)\)