Question

What's the relationship among max and min, concave direction and the derivatives?

Answer 1

Ok
1st derivative:
If f'(x) is positive, f(x) is increasing.
if f'(x) is negative, f(x) is decreasing.
If f''(x) is positive, f(x) is concave up
If f''(x) is negative, f(x) is concave down
When f'(x) = 0, it can be either a min or max. If it goes from increasing to decreasing (+-> -) then it is concave down and a max.
If it goes from decreasing to increasing, then it is concave up and a min.