Suppose is twice differentiable and concave up on

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

Problem

Suppose ff is twice differentiable and concave up on [a,b][a,b]. Let TnT_n be the approximation of the average value of ff on [a,b][a,b] obtained by first estimating ∫abf(x) dx\displaystyle\int_a^b f(x)\,dx with the Trapezoidal Rule using nn equal subintervals and then dividing by b−ab-a. Which of the following best describes the sign of TnT_n minus the exact average value? (a) Negative, because the trapezoids lie below the curve on a concave-up interval. (b) Positive, because the trapezoids lie below the curve on a concave-up interval. (c) Positive, because the trapezoids lie above the curve on a concave-up interval. (d) Zero, because the trapezoidal estimate is exact for concave-up functions.

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