record: 42 score: 42 the graph of the function f is shown below. at which of the following points is f(x)>0…

Explanation:

Step1: Recall derivative - slope relationship

$f'(x)>0$ means the function is increasing. The graph is increasing when the slope of the tangent line is positive.

Step2: Recall second - derivative - concavity relationship

$f''(x)<0$ means the function is concave down. A function is concave down when the slope of the tangent line is decreasing.

Step3: Analyze point a

At point a, the function is decreasing ($f'(x)<0$), so it does not meet the criteria.

Step4: Analyze point b

At point b, the function is increasing ($f'(x)>0$), but it is concave up ($f''(x)>0$) since the slope of the tangent line is increasing.

Step5: Analyze point c

At point c, the function has a horizontal tangent ($f'(x) = 0$), so it does not meet the criteria.

Step6: Analyze point d

At point d, the function is increasing ($f'(x)>0$) and is concave down ($f''(x)<0$) as the slope of the tangent line is positive but decreasing.

Answer:

d