Question

Determine whether the Mean Value theorem can be applied to f on the closed interval [a, b]. (Select all that apply.) f(x) = 9x3, [1, 2] Yes, the Mean Value Theorem can be applied. No, because f is not continuous on the closed interval [a, b]. No, because fis not differentiable in the open interval (a, b). None of the above.

Answer 1

Yes, the Mean Value Theorem can be applied

Explanation:

The Mean Value Theorem states that for a function f which is continuous over the interval [a,b] and also differentiable on the open interval (a,b), there exists a point "c" in the interval (a,b) such that f'(c) is equal to the function's average rate of change over [a,b].

So, we'll need to determine the average rate of change first over [1,2]:

`\displaystyle (f(b)-f(a))/(b-a)=(f(2)-f(1))/(2-1)=9(2)^3-9(1)^3=9(8)-9=72-9=63`

Now, we know that f'(c) = 27c², so we need to solve 27c² = 63 and see if that value of "c" is contained within the open interval (1,2):

`\displaystyle 27c^2=63\\\\c^2=(63)/(27)\\\\c^2=(7)/(3)\\\\c=\sqrt{(7)/(3)}\approx1.528`

Hence, because c = √(7/3) is within the interval (1,2), we can conclude that the Mean Value Theorem can be applied.