Question

7 fill in the blank 4 points
the figure shows the graph of ( f(x) ) on the interval ( -2,4 ) with inflection points at ( x = 0 ) and ( x = 2 ).
the interval(s) where ( f(x)>0 ):
the interval(s) where ( f(x)>0 ):
(use interval notation)

Explanation:

Step1: Determine where \(f'(x)>0\)

The first derivative \(f'(x)>0\) when the function \(f(x)\) is increasing. Looking at the graph, the function \(f(x)\) is increasing on the interval \((-1,3)\).

Step2: Determine where \(f''(x)>0\)

The second derivative \(f''(x)>0\) when the function \(f(x)\) is concave - up. Since the inflection points are at \(x = 0\) and \(x=2\), the function \(f(x)\) is concave - up on the interval \((-2,0)\cup(2,4)\)

Answer:

• The interval where \(f'(x)>0\): \((-1,3)\)
• The interval where \(f''(x)>0\): \((-2,0)\cup(2,4)\)