Iam wondering if the answer to this question is table b. at x = 1, f'(x) = 1/2. at x = 2, f'(x) = 3. does this mean that, between 1 and 2, the slope should be greater than 1/2 and less than 3? (e. g. graph b)?

I agree. The answer is choice B



f ' (1) is shown to be positive
f ' (2) is shown to be positive as well
There are no critical values between x = 1 and x = 2, so f ' (x) is positive on the interval 1 < x < 2, so f(x) is increasing on this interval

All of this points to either choice A or choice B



Similarly, 
f ' (3) is negative
f ' (4) is negative
so f ' (x) is negative where 3 < x < 4 due to no other critical values being between x = 3 and x = 4

Based on these facts alone, the answer is either choice B or choice D

But we know that it can't be choice D as we determined it was between A and B. This rules out choice D

So all that's left is choice B

The answer is 9.7 cm