The Function f(x) from the graph f '(x)

1. The function f(x) is increasing in the interval (-2,0) U (0,2), since the graph is showing the derivative of f(x) when f(x) has a positive slope f(x) is increasing.

The function f(x) is decreasing in the interval (-infinity,-2) U (2, infinity), because a function is decreasing when the slope is negative.

2. There is local minimum at x=-2. Extremas can be found in critical points when f '(x)=0 or undefined. x=-2 is a local minimum because the slope of f(x) changes from being negative to positive.

There is local maximum at x=2 because at this point the slope of f(x) changes from being positive to negative.

3. The function is concave up in the interval (-infinity, -5/4) U (0,5/4). A function is concave up where f ''(x)>0.
The function is concave down in the interval (-5/4,0) U (5/4, infinity). A function is concave down where f ''(x)<0.
I can tell from the graph f ' (x) where f ''(x) is positive and negative by looking where the slope is positive and where the slope is negative since f ''(x) is derivative of f ' (x).

4. f(x) is a power function of 5 because the graph of f '(x) is to the power of 4 so its anti-derivative has to have a power of five.