1

-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]

### 1 Answer by Expert Tutors

John M. | John - Algebra TutorJohn - Algebra Tutor
0
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