What can be said about $f''$ if the trapezoidal approximation is always an overestimate?

Question

For any $a$ and $b$ the Trapezoidal approximation of the integral $\int_a^b f(x)\,dx$ is an overestimate. What can you conclude about the second derivative of $f$?

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I think it might mean that the second derivative is increasing, which would cause the original function to be concave up?

Answer 1

that means the graph of $y = f(x)$ must be below every secant line connecting the points $(a, f(a))$ and $(b, f(b))$ this is one of the characterization of $f$ being concave up. another one is having $f'' > 0.$

in fact you can drive $$\int_a^b f(x) dx = (b-a)\left(\frac12 f(a) + \frac12 f(b)\right) - \frac1{12}(b-a)^3f''(c) \text{for some } c, a < c < b.$$