Final answer:
The student's question about finding the number c for f(x) = ln(x¹⁶) that satisfies Rolle's Theorem cannot be resolved as such a c does not exist within the interval (1, e¹⁵); the function does not meet all the conditions of Rolle's Theorem in this interval.
Step-by-step explanation:
The student is asking how to find a number c in the interval (1, e¹⁵) that satisfies Rolle's Theorem for the given function f(x) = ln(x¹⁶). To apply Rolle's Theorem, f(x) has to be continuous on [1, e¹⁵] and differentiable on (1, e¹⁵), and f(1) must equal f(e¹⁵). It's clear that f(1) = ln(1¹⁶) = 0 and f(e¹⁵) = ln((e¹⁵)¹⁶) = 15ln(e) = 15, since ln(e) = 1. Therefore, f(1) = f(e¹⁵), satisfying the theorem's conditions. We then find the derivative f'(x) = d/dx[ ln(x¹⁶) ] = 16/x. Setting f'(x) to 0 to find c we have 16/c = 0, which does not have a solution for c within (1, e¹⁵), thus such a c does not exist and Rolle's Theorem does not apply here as the function does not meet all the required conditions.