Question

question 2 (1 point)
below is the graph of the derivative of f(x):
on which of the following intervals is f(x) decreasing (select all that apply).
remember: the graph given in the image is the graph of f(x)... it is not the graph of f(x).
(0,2)
(2,3)
(3,4)
(4,5)
(5,6)
(6,8)

Explanation:

Step1: Recall the relationship between \(f(x)\) and \(f^{\prime}(x)\)

A function \(y = f(x)\) is decreasing on an interval \((a,b)\) if \(f^{\prime}(x)<0\) for all \(x\in(a,b)\)

Step2: Analyze the sign of \(f^{\prime}(x)\) on each interval

• For the interval \((0,2)\):

We observe that the graph of \(y = f^{\prime}(x)\) is below the \(x -\)axis (i.e., \(f^{\prime}(x)<0\)) on the interval \((0,2)\)

• For the interval \((2,3)\):

The graph of \(y = f^{\prime}(x)\) is above the \(x -\)axis (i.e., \(f^{\prime}(x)>0\)) on the interval \((2,3)\)

• For the interval \((3,4)\):

The graph of \(y = f^{\prime}(x)\) is above the \(x -\)axis (i.e., \(f^{\prime}(x)>0\)) on the interval \((3,4)\)

• For the interval \((4,5)\):

The graph of \(y = f^{\prime}(x)\) is above the \(x -\)axis (i.e., \(f^{\prime}(x)>0\)) on the interval \((4,5)\)

• For the interval \((5,6)\):

The graph of \(y = f^{\prime}(x)\) is below the \(x -\)axis (i.e., \(f^{\prime}(x)<0\)) on the interval \((5,6)\)

• For the interval \((6,8)\):

The graph of \(y = f^{\prime}(x)\) is above the \(x -\)axis (i.e., \(f^{\prime}(x)>0\)) on the interval \((6,8)\)

Answer:

\((0,2)\), \((5,6)\)