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.

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

Here is the MVT (for integrals):

∫(a to b) f(x) = f(c) (b - a) (for some c ∈ [a,b], provided f(x) is integrable and continuous on the given interval)

Since this function is integrable and continuous on [0, √3], the MVT applies here, and we can write out the following:

∫(0 to √3) 3 - x² = f(c) (√3 - 0) (for some c ∈ [0, √3])

∫(0 to √3) 3 - x² = 3 - c² (√3) (just wrote what f(c) was out in full and simplified second term on the right side)

[ (0 to √3) 3x - x³/3 = 3√3 - √3c²

3√3 - 3√3/3 = 3√3 - √3 c²

-√3 = -√3 c²

c² = 1, or c = ±1

This is d on the list of possible answers.