A differentiable function is strictly increasing and strictly concave down on

MATHEMATICS · Multi-part · A Level · Mathematics · Integration · Integration

Problem

A differentiable function

p

is strictly increasing and strictly concave down on

[1,7]

. The exact values

p(1)=2

and

p(7)=10

are known, but no closed-form expression for

p

is available. Let

L_{6},\,R_{6},\,M_{6},

and

T_{6}

denote, respectively, the left-endpoint, right-endpoint, midpoint, and trapezoidal approximations that use six equal subintervals of

[1,7]

. (a) Order the four approximations

L_{6},\,R_{6},\,M_{6},

and

T_{6}

from least to greatest. Provide a clear justification rooted in the monotonicity and concavity of

p

. (b) Without performing any numerical integration, describe whether

M_{6}

is closer to

\displaystyle\int_{1}^{7}p(x)\,dx

than

T_{6}

. Support your answer with a qualitative error analysis based on the geometry of a concave-down curve.