Question

Does the function `f(x) = ln x` satisfy the hypotheses of the Mean Value Theorem on the given interval [1, 7]?

Answer 1

Is it continuous on the interval `[1,7]`? Yes, the `ln` is continuous on its entire domain, `(0,oo)`

Is it differentiable on the interval `(1,7)`? Yes, `f'(x) = 1/xx` exists for all `x > 0`, so it certainly exists for `x in (1,7)`.

That's it. Yes, the function satisfies the hypotheses of the Mean Value Theorem.