Mean Value Theorem

Question

Determine if the Mean Value Theorem for Integrals applies to the function f(x) = x³ - 4x on the interval [-1, 1]. If so, find the x-coordinates of the point(s) guaranteed to exist by the theorem

Answer 1

Since f(x) = x³-4x is continuous on [-1,1], the theorem does apply.

 

So, by the Mean Value Theorem for integrals, there is a number, c, in the interval (-1,1) such that f(c)(1-(-1)) = ∫_{(from -1 to 1)}f(x)dx             

 

2f(c) = ∫_{(from -1 to 1)}(x³-4x)dx

 

2f(c) = [(1/4)x⁴ - 2x²]_{(from -1 to 1)}

 

2f(c) = (-1.75 - (-1.75)

 

  c³-4c = 0

 

  c(c²-4) = 0

 

  c = 0, 2, -2

 

The only value of c that is in the interval (-1,1) is c = 0.