Question

Given the function `f(x)=(x-4)^2-1`, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [3,0] and find the c?

Answer 1

`c=3/2`

Explanation:

Mean value theorem states that for a function defined and continuous on `[a,b]` and continuously differentiable on `(a,b)` then there exists some `a < c < b` such that

`f'(c) = (f(b)-f(a))/(b-a)`

With `a = 0 and b = 3` we have

`(((3-4)^2-1) - ((0-4)^2-1))/3 = -5`

`f'(x) = 2(x-4)`

`implies f'(c) = 2(c-4)`

`2c - 8 = -5`

`2c = 3 implies c = 3/2`