A twice–differentiable function satisfies for all in

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

Problem

A twice–differentiable function ff satisfies ∣f′′(x)∣≤3|f''(x)|\le 3 for all xx in [0,2][0,2]. The values f(0)=4f(0)=4, f(1)=3f(1)=3, and f(2)=1f(2)=1 are known. An approximation of ∫02f(x) dx\displaystyle\int_{0}^{2} f(x)\,dx is obtained by connecting the data points with straight‐line segments and finding the enclosed area. What is the greatest possible error in this approximation? (a) 0.250.25 (b) 1.001.00 (c) 0.500.50 (d) 0.1250.125

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