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Darboux's theorem states that if a function has a derivative at every point in an interval, then the derivative satisfies the Intermediate Value Property, meaning that the function takes on all values between f'(a) and f'(b) in the interval (a, b).

In our case, we are given that f(x) is differentiable on R, so it has a derivative at every point in the interval [1, 2]. By Darboux's theorem, f'(x) satisfies the Intermediate Value Property on this interval. Therefore, since f'(1) = 5 and f'(2) = 6, for any value k in the interval [5, 6], there must exist some x in the interval [1, 2] such that f'(x) = k.

Thus, we can conclude that
[5,6] ⊆ {f'(x)|1≤x≤2}.