Question

let f(x)=x^3+1, show that x satisfies the hypothesis for the minimum value theorem on the interval [1,2] and find the values of c in this interval

Answer 1

Is it meant to minimum value or mean value...? For mean value, it needs to be continuous on [a,b] (which is [1,2] here) and differentiable on (a,b) and c comes from... \[\large f'(c) = \frac{ f(b) - f(a) }{ b - a }\]
So first you need to find the first derivative, f'(x), then get f(2) and f(1), and plug them all in.