`f(x) = sqrt(x), [0,4]` Find the number `c` that satisfies the conclusion of the Mean Value Theorem on the given interval.
For the mean value theorem to be valid, the function `f(x) = sqrt x` must satisfy the following conditions on the interval, such that:
f(x) is continuous over the interval [0,4] and it is because it is an elementary function.
f(x) is differentiable on (0,4).
If both conditions are satisfied, then, it exists a point `c in (0,4)` , such that:
`f(4) - f(0) = f'(c)(4 - 0)`
You need to evaluate f(4) and f(0), by replacing 4 and 0 for x in equation of the function:
`sqrt 4 - sqrt 0 = (sqrt c)'(4-0)`
2 - 0 = 4/(2sqrt c)
Reducing by 2 yields:
`2 = 2/sqrt c => 2sqrt c = 2 => sqrt c = 1 => c = 1 in (0,4)`
Hence, evaluating the number c that satisfies the mean value theorem yields c = 1.