Question

Suppose function f(x) is twice differentiable on [a, b], and

suppose that its second derivative f''(x) is bounded and Riemann
integrable on [a, b]. Use Lebesgue’s Criterion for Riemann
Integrability

Answer 1

The question pertains to verifying if a twice differentiable function with a bounded and Riemann integrable second derivative can be integrated on an interval [a, b], using Lebesgue's Criterion for Riemann Integrability.

Step-by-step explanation:

The question involves applying Lebesgue's Criterion for Riemann Integrability to a twice differentiable function whose second derivative is bounded and Riemann integrable. Lebesgue's Criterion states that a function is Riemann integrable if and only if it is bounded and its set of discontinuities has measure zero. Since the second derivative of the function, f''(x), is bounded and Riemann integrable on the interval [a, b], it satisfies the criteria for Lebesgue's Criterion. This means that the function f(x) can be integrated using Riemann integration over the interval [a, b].