The Second Derivative, Local Maxima and Minima, and Inflection Points

Its really cool to think that f",f', and f are related. f" lets us know depending on whether it is less than or greater than zero, which describes if f' is concave up or down! I enjoy clear connections between ideas. However I did not understand the government defense budget example, perhaps that was worded funny?

Local Maxima and Minimum were not too hard of an idea to grasp, but one clarification is the critical point f'(p)=0 reflected in the original function as crossing an axis? I want to make sure I am drawing the correct conclusions from the reading. I had the most difficulty with the idea that critical points could divide the domain of f into intervals. If there are critical points, must we look at the function f in sections defined between two critical points or can it still be looked at as a whole?

The idea of an inflection point was pretty easy, its just when the concavity changes, but its neat to realize inflections occur when f" changes signs.

I think the class will have the most difficulty with keeping all the relationships together. Also perhaps the units of the second derivative. Understanding f" as a rate of change, I'm not quite sure if it is as simple if the second derivative is positive then the rate of change is increasing and vice versa. It seems to simple to be right... Or am I over thinking it?

The mid-term exhausted mind is a funny thing!