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

1. The function f(x) is increasing at (-2,0) and (0,2) and decreasing at (negative infinity,-2) and (2, infinity). A function is increasing when f '(x)>0 and decreasing when f '(x)<0. Because the given graph is that of f '(x), we only have to check the output of the graph to find out when f '(x) is positive and negative. It can be told from looking at the graph that f '(x) is positive at (-2,0) and (0,2), meaning the function f(x) is increasing at those intervals. Similarly, it can be told that f '(x) is negative at (negative infinity, -2) and (2, infinity), so the function f(x) is decreasing at those intervals.

2. There is a local minimum at x=-2 and a local maximum at x=2. Extrema can only occur at critical points, or when f '(x)=0 or is undefined. In this case, from looking at the graph, those points would be at x=-2, x=0, and x=2. For there to be an extrema, the sign of f '(x) must change. Because it fails to do so at x=0, it isn't an extrema. At x=-2, the sign of f '(x) changes from negative to positive, so it must be a local minimum. At x=2, the sign of f '(x) changes from positive to negative, so it must be a local maximum.

3. The function f(x) is concave up at (negative infinity,-1.5) and (0, 1.5) and is concave down at (-1.5, 0) and (1.5, infinity). (Note: the values -1.5 and 1.5 are approximations, not the exact values). A function is concave up when f "(x)>0 and concave down when f "(x)<0.>0) and is concave down when the graph of f '(x) is decreasing (f "(x)<0). Just by looking at the graph, f '(x) is increasing at (negative infinity, -1.5) and (0, 1.5), so that is when the graph of f(x) is concave up. In the same manner, f '(x) is decreasing at (-1.5, 0) and (1.5, infinity), so that is when the graph of f(x) is concave down.

4. I would say f(x) is a fifth power polynomial equation. Normally, the graph of a derivative function is one degree less that the original function. The derivative of looks as if it is fourth power polynomial equation, so one can guess f(x) is a fifth power polynomial equation.