Main Points

Because f'(x) is function, evident by its graph, it too has a deriv. f"(x), like any deriv, shows certain properties for f'(x), which in turn indicates certain things for f(x).  When f" is positive, the function f' is increasing, therefore f is concave up. The opposite is true for when f" is negative. The local maximum is the highest value on the part or the curve, while the local minimum is the lowest. A critical value is a point where the deriv is undefined or =0. It is also the maximum or minimum of a function. For a function f, the min occurs when a function changes at p when the function changes from decreasing to increasing. When the change is form increasing to decreasing, it is a max. Because concavity is related to a function increasing or decreasing, when x is concave down there is local max, when it concaves up there is a local min.

Changes in convavity result in inflection points.

Challenges. 

How can p be a point. Isn't a point an x and y value, there for how can there be f(p)?

Reflection.

I have gone over maximum and minimums and inflection points in my previous calc course.  I think the better understanding of derivs i now have allows me to understand the implications of concavity and therefore max and mins, because they occur at points of concavity.