A twice-differentiable function satisfies on and on

MATHEMATICS · Single correct · A Level · Mathematics · Integration · Integration

Problem

A twice-differentiable function

p

satisfies

p''(x)>0

(0,1)\cup(3,4)

and

p''(x)<0

(1,3)

. The interval

[0,4]

is partitioned into four subintervals of equal length, and the midpoint approximation

P_M

and trapezoidal approximation

P_T

\displaystyle \int_{0}^{4} p(x)\,dx

are computed using that partition. Which of the following statements is necessarily true? (a)

\displaystyle P_M<\int_{0}^{4} p(x)\,dx<P_T

(b)

\displaystyle \left|P_M-\int_{0}^{4} p(x)\,dx\right|<\left|P_T-\int_{0}^{4} p(x)\,dx\right|

(c)

\displaystyle P_T<\int_{0}^{4} p(x)\,dx<P_M

(d)

\displaystyle \left|P_M-\int_{0}^{4} p(x)\,dx\right|=\left|P_T-\int_{0}^{4} p(x)\,dx\right|