To begin with the table, if I am interpreting it correctly, the first derivative is slope. So the curve has positive slope to x=1, zero slope at 1 and negative slope after 1. This describe a function maximum.
The second derivative is concavity. Inflection points are places where rate of curviture changes from positive to negative. In this case, there are 2. At 1 and +6.
The lettered items further describe the curve. The whole domain in x described is 3 to +7. Putting the descriptions in left to right xorder the curve is:
concave up 3 to 1.5
concave down 1/2 to 4
f(x) has a maximum at 1
the curve changes it's concavity twice, at 1.5 and 4
Putting the entire set together, f(x) is a curve that increases from 3 to 1, where it reaches a maximum, and decreases to +7. The curve is concave up from 3 to 1.5, constant slope to .5, concave down to 4.
It's difficult to graph in this interface but it should look something like:
x
x x
x x
x x
x x
x
x
x
3 .5 4 7
12/1/2013

John M.
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