`f(x) = (x + 1)/x, [-1,2]` 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 1 for b and 0 for a, yields:
`f(2) - f(-1) = f'(c)(2+1)`
Evaluating f(2) and f(-1) yields:
`f(2) =(2+1)/2 => f(2) =3/2`
`f(-1) = (-1+1)/(-1) => f(-1) = 0`
You need to evaluate f'(c), using quotient rule:
`f'(c) = ((c+1)/c)' => f'(c) = ((c+1)'*c - (c+1)*c')/(c^2) => f'(c) = (c - c - 1)/(c^2) => f'(c) = -1/(c^2)`
Replacing the found values in equation `f(2) - f(-1) = f'(c)(2+1):`
`3/2-0= -3/(c^2) =>1/2 =-1/(c^2)=>c^2 = -2 !in R`
Hence, in this case, there is no valid c value for the mean value theorem to be applied.