Mean Value Theorem for Integrals

Question

Determine if the Mean Value Theorem for Integrals applies to the function f(x) = 3 - x² on the interval [0,√3]. If so, find the x-coordinates of the point(s) guaranteed by the theorem.

Answer 1

Since f(x) = 3-x² is continuous on the interval [0, √3], the MVT does apply.

 

The MVT says that there is a number, c, in the interval so that:

 

              f(c) = 1/(√3 - 0)∫_{(from 0 to √3)}(3-x²)dx

 

              f(c) = 1/√3 [3x - (1/3)x³]_{(from 0 to √3)}

 

              f(c) = 1/√3[3√3 - √3]

 

              f(c) = 2

 

So, 3 - c² = 2

 

           c² = 1      c = 1