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Calculus Question.Please help!!

What can you conclude about the graph of f(x) from the following table? justify your answer
 
                       -1                       +6
f'(x)        +        0           -           -            -
f''(x)       +        0           -           0            -
 
a) f(x) is concave down on [-1/2,4] and f(7)<f(5.1)
b)x=6 is an inflection point and f(-2)>f(5.1)
c)x=-1 is an inflection point and f(-0.3)<f(-0.2)
d)f(x) is concave up on [-3,-1.5] and f(x) is increasing on [-2,-1/2]
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1 Answer

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 x-order 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
 

Comments

"The lettered items further describe the curve." John, When I read this question, I am not convinced that the lettered items describe the curve. It appears that they may be choices of an answer to the question. On what basis did you decide that this statement was true?

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