Let be twice differentiable on with for all in

MATHEMATICS · Subjective · A Level · Mathematics · Integration · Integration

Problem

Let

f

be twice differentiable on

[a,b]

with

f''(x)\ge 2

for all

x

(a,b)

. Using

n

equal subintervals, let

M_n

and

T_n

denote the midpoint and trapezoidal approximations, respectively, of

\displaystyle \int_{a}^{b} f(x)\,dx

. Determine the sign (positive, negative, or zero) of the errors

E_M=M_n-\int_{a}^{b} f(x)\,dx

and

E_T=T_n-\int_{a}^{b} f(x)\,dx

. Then state which error has the smaller absolute value. Support your conclusions with reasoning based on the given information about

f''