F'(x) graph

1. Where is the function, f(x), increasing? Where is it decreasing? How can you tell from this graph? Explain.

-f(x) is increasing when f'(x)>0
-f(x) is decreasing when f'(x)<0>

Since this graph is f'(x), it graphs the slope of f(x). As a result, The function is increasing(positive slope) between (-2,0)U(0,2). The function is decreasing(negative slope) between (-∞,-2)U(2,∞).

2. Where is there an extrema? Explain. (There are no endpoints.)

-There's a possibility of having "critical points" when f'(x)=0

• There is a local minimum at x=-2, because the slope=0 and it changes from negative to positive.
  
• There is a local maximum at x=2, because the slope=0 and it changes from positive to negative.
• There isn't a critical point at zero, because there isn't a change of signs
  

3. Where is the function, f(x), concave up? Where is it concave down? How can you tell from this graph?

-f(x) is concave up when f''(x)>0
-f(x) is concave down when f''(x)<0

• Concave up:(-∞,-1)U(0,1)
• Concave down:(-1,0)U(1,∞)

4. Sketch the graph f(x) on a sheet of paper. Which power function could it be? Explain your reasoning.

• If you were to find the anti derivative of f'(x), f(x) would be an x^5 power function, because f'(x) has 4 slope changes.