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

Avarage value of function f_av=1/(sqrt3 - 0)·∫₀^{3^1/2}(3 - x²)dx = 1/sqrt3 ·(3x - x³/3)₀^sqrt3 = 3 - 1 = 2.

To get number c such that f(c) = f_av we have equation: 3 - c² = 2, c² = 1 and c = 1, because c from interval 0 and sqrt3.

Answer: f_av = 2 and c = 1