1. Where is the function, f(x), increasing? Where is it decreasing? How can you tell from this graph? Explain.
- f(x) is increasing between (-2,0)U(0,2). This is due to the fact that when f '(x)>0, the graph of f(x) is increasing.
- f(x) is decreasing from (- infinity, -2)U(2, infinity). I can tell from this graph because whenever f '(x)< 0, it means that the graph of f(x) is decreasing.
- (On the graph f '(x), anything above the x axis is where the graph f(x) is increasing and anything below the x axis is where f(x) is decreasing. **Remember that f '(x) graphs the slope of f(x).)
2. Where is there an extrema? Explain. (There are no endpoints.)
If you're talking about the extrema on the graph f(x)....
- Local Minimum: x=-2 (-2,0) A Local minimum occurs at the point where f ' (x)< 0 and then changes to f ' (x)>0, also known as a critical point ( f ' (x)=0 or undefined ) [the function has to change from negative to positive slope in order for it to be a local minimum).
- Local Maximum: x=2 (2,0) A Local maximum occurs at the point where f ' (x)>0 and then changes to f ' (x)<0, which is also a critical point. (the function has to change from positive slope to negative slope in order for it to be a local maximum).
(Way clearer explanation than on my test, >:\)
3. Where is the function, f(x). concave up? Where is it concave down? How can you tell from this graph?
- Concave up: (-infinity, -1.25)U(1.25, infinity). The function is concave up whenever f "(x)>0
- Concave down: (-1.25, 0)U(0, 1.25). The function is concave down whenever 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 f(x) appears to be an x^5. Since the graph of f ' (x) looks like an x^4, my prediction is reasonable since an x^4 is the derivative of an x^5 graph.
