Question

Determine if the Mean Value Theorem for Integrals applies to the function f(x) = 3 - x^2 on the interval the closed interval from (0, sqrt(3) . If so, find the x-coordinates of the point(s) guaranteed by the theorem.

Answer 1

I said Yes, x = 1

Answer 2

\[\frac{ 1 }{ \sqrt{3} }\int\limits_{0}^{\sqrt{3}}3-x^2\]

Answer 3

eh, im not sure if thats right sorry. Thats for the average

Answer 4

thats ok but i got 1/6

Answer 5

Before you do anything, you need to check that \(f\) satisfies the conditions for the MVT - it needs to be continuous on the interval. Not a problem in this case, as \(f\) is a polynomial, and polynomials are continuous everywhere.

The MVT states that there is some \(c\in(0,\sqrt3)\) such that
\[\int_0^{\sqrt3}\left(3-x^2\right)\,\mathrm{d}x=f(c)(\sqrt3-0)=\sqrt3(3-c^2)\]Integrate the left side and solve for \(c\).