From f(x) to f'(x)

1.) Where is the function, f(x), increasing? Where is it decreasing? How can you tell from this graph? Explain
The function f(x) is increasing in the intervals (-2,0) U (0,2) because there all the outputs for f '(x) are positive, which means that f(x) had a positive slope in all those x values at that interval and therefore it was increasing. The graph of f(x) is decreasing in the intervals: (-infinity, -2) U (2, infinity), because at those intervals the outputs for f '(x) are negative and therefore f(x) had a negative slope while it decreased.

2.) Where is there an extrema? Explain. (There are no endpoints.)
There is an extrema at x= -2 , 0, 2
There is an extrema at those values of x because there f '(x) is zero and since wherever f(x) has a slope of its tangen line (f '(x)) equals to zero this means that the graph has reached a plateau, which is like a high stable elevation. And because f '(x) is the derivative/slope from f(x) , at that plateau of f(x) the slope it's going to be zero. Also we can know the extrema by using the critical points, which happen when f ' (x) = to zero or it's undefined or at endpoints. However, we don' t have endpoints but we can see that f '(x) =0 at x= -2, 0, 2 as said before.

3.) Where is the function, f(x), concave up? Where is it concave down? How can you tell from this graph?
The function f(x) is concave up at the intervals (-infinity, -1.20) U (0, 1.20) because there is where f ' (x) has a positive slope and it's decreasing and we know by our concavity test that f(x) it's going to be concave up when f ''(x), which is the derivative of f ' (x), it's greater than zero (f ''(x)>0). This means that it's going to be concave up when f ' (x) it's increasing. The graph it's going to be concave down in the intervals (-1.20, 0) U (1.20, infinity) which is where f ' it's decreasing. And when f ' (x) is decreasing, f ''(x) it's going to be negative ( f ''(x)<0)>

4.) Sketch the graph f(x) on a sheet of paper. Which power function could it be? Explain your reasoning.
The graph should look like an f(x)= x^5 since f ' (x) it's x^4 , and according to the pattern if you go from f(x) to f ' (x) the latter will drop down a degree but if we go from f ' (x) to f(x) then I would add one degree to find f(x).