Question

Determine if the Mean Value Theorem for Integrals applies to the function f of x equals the square root of x on the interval [0, 4]. If so, find the x-coordinates of the point(s) guaranteed to exist by the theorem.

Answer 1

check that f(x) satisfies the theorem's conditions:
yes, f(x) is continuous on [0,4] so we can go on.

the MVT states that there is some c in [0,4] such that
f(c) = 1/(4-0)*∫[0,4] f(x) dx

I get c = 4/3, which is indeed in [0,4]

Answer 2

x = 16/9
8/3 = 2sqrt(x)
4/3 = sqrt(x)
16/9 = x

Answer 3

it's not 4/3 but you're close Steve, you just needed to do one more step