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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John M. | John - Algebra TutorJohn - Algebra Tutor

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

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