Question

Can the mean value theorem be applied to `f(x) = 2(sqrt x) + x` on the interval [1,4]?

Answer 1

Yes it can.

Hypothesis 1:
Is `f` continuous on the closed interval `[1, 4]`?

Yes, it is continuous on its domain (`x>=0`.)

Hypothesis 2
Is `f` differentiable on the open interval `(1, 4)`?

Yes. `f'(x) = 1/sqrtx +1` exists for all `x>0`, so it exists in `(1, 4)`.

That's all that is required to apply the Mean Value Theorem.