**8. Below is a graph of \( f' \) (DERIVATIVE). Assume that the domain is all real numbers.**---**Graph Explanation:**- **Axes:** - The horizontal axis (x-axis) ranges from \(-1\) to \(4\). - The vertical axis (y-axis) ranges from \(-4\) to \(2\).- **Curve Details:** - The graph represents a smooth curve of the derivative \( f' \). - The curve intersects the y-axis above the point \(1\), indicating that \( f’(0) \) is positive. - It crosses the x-axis near \(x = 0.5\), suggesting a critical point. - It reaches a local maximum above \(x = 1\). - The graph then descends, crossing the x-axis just after \(x = 2\). - It continues downward, reaching a local minimum near \(x = 3\) before rising back towards the x-axis.This graph is used to analyze the behavior of the function \( f \) by examining the qualities of its derivative \( f' \). Understanding this graph helps in identifying intervals of increase or decrease in \( f \), as well as locating critical points and inflection points. (d) Estimate the x-values of the inflection points of \( f \).(e) On what intervals is \( f \) concave up? concave down?(f) Using the information above and assuming that \( f(0) = -2 \), sketch a graph of \( f \) on top of the graph of \( f'' \) below.**Graph Explanation:**The graph shown is of \( f'' \) (the second derivative of \( f \)). It features a curve plotted against the x-axis ranging from approximately -1 to 4. Key features include:- A crossing of the x-axis at around \( x = 0 \) and \( x = 2.5 \).- The graph is above the x-axis around \( x = 0 \) to \( x = 2.5 \), indicating intervals where \( f \) is concave up.- The graph dips below the x-axis approximately beyond \( x = 2.5 \), indicating an interval where \( f \) is concave down.These characteristics will help identify the intervals of concavity and estimate the inflection points for the function \( f \).

d) The second derivative of the function value are either zero, or, not defined for x-values at inflection points. Thus, the graph where f'(x) has horizontal tangent will be the inflection point for f(x). Therefore, we get x=0, x=1, x=3. e) The function is concave up when f''(x)>0, so, we must have (f'(x))'>0, or, f'(x) must be increasing. Thus, the intervals are 0,1, 3, ∞. Similarly, the function is concave down when f''(x)<0, so, we must have (f'(x))'<0, or, f'(x) must be decreasing. Thus, the intervals are -∞, 0, 1, 3. f) The maximum or minimum occurs when f'(x)=0. The function is increasing if f'(x)>0 while the function is decreasing if f'(x)<0. Thus, the required graph with f(0)=-2 and other information is as follows: