`f(x) = 2x^3, [0,6]` Determine whether the Mean Value Theorem can be applied to `f` on the closed interval `[a,b]`. If the Mean Value Theorem can be applied, find all values of `c` in the open interval `(a,b)` such that `f'(c) = (f(b) - f(a))/(b - a)`. If the Mean Value Theorem cannot be applied, explain why not.
The mean value theorem is applicable to the given function, since it is a polynomial function. All polynomial functions are continuous and differentiable on R, hence, the given function is continuous and differentiable on interval.
The mean value theorem states:
f(b) - f(a) = f'(c)(b-a)
Replacing 6 for b and 0 for a, yields:
`f(6) - f(0) = f'(c)(6 - 0)`
Evaluating f(6) and f(0) yields:
`f(6) = 2*6^3 => f(6) = 2*216 = 432`
`f(0) = 0`
You need to evaluate f'(c):
`f'(c) = (2c^3)' => f'(c) = 6c^2`
Replacing the found values in equation `f(6) - f(0) = f'(c)(6 - 0):`
`432 - 0 = 6c^2(6 - 0) => 36c^2 = 432 => c^2 = 12 => c = sqrt12 => c = 2sqrt3`
Hence, in this case, the mean value theorem can be applied and the value of c is `c = 2sqrt3.`