What is f '(x) saying about f(x)

This is the graph f '(x) which simply means that what you see on this graph are the slopes of the graph f(x). So what is this graph telling us about our invisible f(x).

1) Where is the functions f(x) increasing? Where is it decreasing? And how exactly do we know?

Ok, so as I mentioned before, this is the graph of the slope of f(x) so where is the slope increasing? F(x) is increasing from (-2,o) U (0, 2). When f '(x) >0 the function f(x) is increasing. -2,0,2 are exclusive because at those points the slope is zero. F(x) is decreasing from (-∞, 2) U (2, ∞) because when f '(x) <0> the function f(x) is decreasing. I a visual learner so for those like me here you go. Hope it helps.

2) Where is there an extrema?

So, to answer this question you must already know that to find an extrema you have to find either the end points or the critical points. Because this graph has no end points we have to find critical points. A critical point is found when f '(x) is either Zero or Undefined. As you can clearly see on the given graph f '(0) is zero so your extrema is at (0,0).

3) Where does f(x) concave up? Where does it concave down?

F(x) is concave up when the slope of f '(x) is positive which is approximately from (-∞, -1.25) U (0, 1.25) and concaves down approximately from (-1.25, 0) U (1.25, ∞)

4) So what does f(x) look like on a graph?

It has to be a x^5 because f '(x) is x^4 and the anti derivative has to reach x to the power of five.