Let be twice differentiable on with and for all in

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

Problem

Let

h

be twice differentiable on

[1,4]

with

h'(x)>0

and

h''(x)<0

for all

x

[1,4]

. For

n\,(\ge 1)

equal-width subintervals, let

L_n,\,R_n,\,M_n,

and

T_n

denote the left-endpoint, right-endpoint, midpoint, and trapezoidal approximations, respectively, of

\displaystyle \int_{1}^{4} h(x)\,dx

. Which inequality is always satisfied? (a)

L_n < M_n < \displaystyle\int_{1}^{4} h(x)\,dx < T_n < R_n

(b)

R_n < T_n < \displaystyle\int_{1}^{4} h(x)\,dx < M_n < L_n

(c)

L_n < T_n < \displaystyle\int_{1}^{4} h(x)\,dx < M_n < R_n

(d)

L_n < T_n < M_n < \displaystyle\int_{1}^{4} h(x)\,dx < R_n