Mean Value Theorem: Find Point [1,4]

[Shadowless]

Homework Statement

State the Mean Value Theorem and find a point which satisfies the conclusions of the Mean Value Theorem for f(x)=(x-1)³ on the interval [1,4].

2. The attempt at a solution

Mean Value Theorem:states that there exists a c∈(a,b) such that f'(c)=[itex]\frac{f(b)-f(a)}{b-a}[/itex]

3(x-1)²=[itex]\frac{27-0}{4-1}[/itex]

3(x-1)²=[itex]\frac{27}{3}[/itex]

3(x-1)²=9

→(x-1)²=3

→x=1±√3

∴x=1+√3 which lies on the interval [1,4].

I was wondering if I did the question correctly and if there was anything further I should add.

 

[voko]

This is all good. Almost. You need to state certain conditions for f(x) and verify that the given f(x) meets them.

 

[Shadowless]

What 'certain condition' are we talking about here?

 

[Infrared]

The hypothesis of the MVT; that f is continuous on the interval and differentiable every in the interior of the interval.