First, I disagree with the answer sheet. That is why I am posting this question here.
The question: Part B, Question 3a.
The answer given (Sorry for the picture being so small).
I know that math is wrong. $\frac{804-1291}{9-7.5} = -324.\bar6$
I still don't like the answer given, because the time to be estimated is 5pm, which is closer to 4:30pm rather than 6pm. Wouldn't the answer be closer to the value for 4:30pm rather than 6pm?
The answer seems to assume the function is piecewise-linear although it is only constrained to be twice-differentiable. It is the rate of change which is being asked for, so if it is piecewise-linear, any time between 4:30 and 6:00 would have the same value (it is negative because the number of people signing in is decreasing between 4:30 and 6:00). But according to the way the question is worded, any value is possible, because for example a polynomial (all are twice-differentiable) can be made to fit.