The function is twice differentiable and concave down on

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

Problem

The function

g

is twice differentiable and concave down on

[1,5]

. Using four equal subintervals, a student obtained a midpoint estimate

M_{4}=10.11

and a trapezoidal estimate

T_{4}=9.63

for

\displaystyle\int_{1}^{5} g(x)\,dx

. Which of the following statements must be true? (a)

M_{4} < T_{4} < \displaystyle\int_{1}^{5} g(x)\,dx

(b)

T_{4} < \displaystyle\int_{1}^{5} g(x)\,dx < M_{4}

(c)

\displaystyle\int_{1}^{5} g(x)\,dx = \dfrac{M_{4}+T_{4}}{2}

(d)

M_{4} < \displaystyle\int_{1}^{5} g(x)\,dx < T_{4}