Suppose is twice differentiable on and satisfies for all in
MATHEMATICS · Single correct · A Level · Mathematics · Integration · Integration
Problem
Suppose f is twice differentiable on (a,b) and satisfies f′′(x)<0 for all x in (a,b). Which of the following statements best explains why the trapezoidal approximation to ∫abf(x)dx obtained with a single subinterval must underestimate the exact value?
(a) For a concave down function, the straight line segment joining the endpoints lies above the graph on (a,b), so the trapezoidal approximation is greater than the exact integral.
(b) Because f′′(x)<0, the function is decreasing on (a,b), and averaging the left and right endpoint values therefore produces too small an estimate.
(c) For a concave down function, the straight line segment joining the endpoints of the graph lies below the graph on (a,b), so the trapezoid’s area is less than the area under the curve.
(d) The trapezoidal rule always underestimates an integral when f′′(x)<0 because each trapezoid’s top edge is horizontal, not slanted like the curve.