Determine if the Mean Value Theorem for Integrals applies to the function f(x) = 3 − x2 on the interval the closed interval from 0 to the square root of 3.

Question

Determine if the Mean Value Theorem for Integrals applies to the function f(x) = 3 − x² on the interval . If so, find the x-coordinates of the point(s) guaranteed by the theorem.

No, the Mean Value Theorem for Integrals does not apply 

Yes, x = 1 

Yes, x = 2 

Yes, 

Answer 1

Hare f(x) = 3-x^2 on [0, √3 ]

It is a polynomial function so it is continous on the closed interval [0, √3 ] and differentiable in (0, √3 )

Thus both condition of Lagrange's mean value theorem are satisfied

Thus there exists c belongs to (0, √3 ) such that

f'(c) = f (√3 )-f(0) ⁄ ( √3 - 0 )

-2c = - √3

c = √3 / 2

since c belongs to ( 0, √3 )

Lagrange's mean value theorem is verfied