Let M_n be the nth Midpoint Rule approximation for a function f.
Let T_n be the nth Trapezoidal Rule approximation.
Show that if f''(x) >= 0 (i.e. f is convex on [a,b]), then for any natural numbers m,n, we have
M_n <= integral f(x)dx <= T_m.
If f''(x) <= 0 on [a,b], this inequality is reversed.
I've been playing around with the respective error bounds forever. How do you make this work algebraically? I'm having trouble simplifying the expression and working out the absolute values.