Determine if the Mean Value Theorem for Integrals applies to the function f of x equals the square root of x minus 1 on the interval [1, 5]. If so, find the x-coordinates of the point(s) guaranteed...

Question

Determine if the Mean Value Theorem for Integrals applies to the function  on the interval [1, 5]. If so, find the x-coordinates of the point(s) guaranteed to exist by the theorem.

A	Yes,  .
B	Yes,  .
C	Yes,  .
D	No, the theorem does not apply.

Answer 1

Yes, the theorem applies because f is continuous on [1,5], there is a point c in(1,5) which has the mean value of f on [a,b]

1/(5-1)∫₁⁵ √(x-1) dx = 1/(5-1)2/3(x-1)^3/2 |₁⁵ = 1/4*2/3(4^3/2 -0^3/2) = 1/4*2/3(8-0) = 4/3

Then, we seek c such that √(c-1) = 4/3

c-1= (4/3)²= 16/9

c = 16/9 + 1

c = 25/9

C