Final answer:
The question concerns evaluating the integral of xf''(x). However, due to missing information regarding f'(0) or the limits of the integral, it is not possible to provide a specific numerical value. Integrating f''(x) would typically yield f'(x), but additional data is needed to fully solve the problem.
Step-by-step explanation:
The student asks to evaluate the integral of xf''(x). To approach this question, we need to consider the properties of integrals and derivatives. The fundamental theorem of calculus connects differentiation and integration, indicating that differentiation can undo the process of integration. If we have a continuous function f(x), then the integral of f'(x) from a to b equals f(b) - f(a).
Given that f is twice differentiable, we can say that integrating f''(x) from 0 to 1 will give us f'(1) - f'(0), and we know that f'(1) = 6. However, since we don’t have the value for f'(0), we cannot directly find the integral of f''(x). More information or additional conditions would be necessary to evaluate the integral of xf''(x). The integral of xf''(x) may be part of an integration by parts process, but without additional context or boundaries for the integral, we cannot give a numerical value.