# `f(x) = sqrt(x), [0,4]` Find the number `c` that satisfies the conclusion of the Mean Value Theorem on the given interval.

### Textbook Question

Chapter 4, 4.2 - Problem 13 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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sciencesolve | Teacher | (Level 3) Educator Emeritus

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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.