I found this question in my exam paper about the trapezoidal rule which I cannot understand the answer.
In the question there is a function that has a graph which closely resembles y=x^2-4x, so I will state my problem using y=x^2-4x.
This is the graph of y=x^2-4x.
https://www.google.com/search?q=y%3Dx%5E2-4x&rct=j
If I use the trapezoidal rule to find the integral for the curve in the range of 0<x<4, I get a negative number as the answer, which is normal since the curve is below the x-axis in that range.
If I find the second derivative of the function, f ' ' (x) > 0, so it proves the curve is pointing upwards, just like the graph shown in the link given.
Since I know that the curve is pointing upwards AND the graph is below the x-axis in that range, I answered that the trapezoidal rule estimate was an under-estimate. (Just like graphs that are concave downwards and above the x-axis)
However the answer guide says that it is an over-estimate because f ' ' (x) > 0.
So is it necessary to take into account the fact that the graph is below the x-axis when I determine whether the estimate is an under-estimate or over-estimate? If not, why?
Or was the answer guide wrong?
Thank you for your guidance.