Are you sure that the first table doesn't list values for and not ? Otherwise the problem does not make sense, because the derivative must either be zero or undefined at a critical point.
Assuming that this is the case:
Originally Posted by
chukie If all the critical values of f(x) are shown below in the chart, which of the following could be values for f'(x) at x=1.5,2.5 and 3.5?
What do you know about critical points?
We know that can change signs only at a critical point and nowhere else (but just because there is a critical point does not mean the derivative must change signs). So can change from increasing to decreasing or from decreasing to increasing only at the critical points.
Plotting the points above, we get something like
Code:
y
^


3+ *

2+

1+
 *
+++++> x
 1 2 3 4
1+

2+

3+ *

4+

5+ *

So is increasing for and decreasing for (remember it can't do both on the same interval, because the change can only happen at a critical point). Thus, must be positive on , and negative on .
So at our table should look something like
Now see which ones work:
Originally Posted by
chukie 1)
This works!
Originally Posted by
chukie 2)
This doesn't!
Originally Posted by
chukie 3)
This doesn't!
Originally Posted by
chukie 4)
Nor does this.
Originally Posted by
chukie 5)
Nope.
Originally Posted by
chukie I am really confused by this question, because for the critical values shown in the chart, isnt f'(x) suppose to be zero at 1, 2, 3 and 4?
Well, must either be zero or undefined at the critical points.