Does the function satisfy the hypotheses of the Mean Value Theorem on the given interval? f(x) = x x + 6 , [1, 12] Yes, it does not matter if f is continuous or differentiable; every function satisfies the Mean Value Theorem. Yes, f is continuous on [1, 12] and differentiable on (1, 12). There is not enough information to verify if this function satisfies the Mean Value Theorem. No, f is not continuous on [1, 12]. No, f is continuous on [1, 12] but not differentiable on (1, 12). Correct: Your answer is correct.

Step-by-step explanation:

the function you have given is not well stated out since it is not division or multiplication. however i will use the function

  and i believe with this example you will get a better view on how to answer similar questions.

for f(x) to satisfy the Mean Value Theorem, it must be continuous in the closed interval [1, 12] and differentiable in the open interval (1, 12).

f(x) is undefined if x= -6 ∉[1, 12]. hence f(x) is continuous on the closed interval [1, 12].

next we check if f(x) is differentiable in the open interval (1 , 12).

using quotient rule, we have

          =

at x=-6, f'(x) is undefined but -6∉( 1 , 12). thus f(x) is differentiable. hence the function satisfies Mean Value Theorem