Question

How do I find the numbers `c` that satisfy the Mean Value Theorem for `f(x)=x^3+x-1` on the interval `[0,3]` ?

Answer 1

The value of `c` is `sqrt{3}`.

Let us look at some details.

M.V.Thm. states that there exists `c` in (0,3) such that

`f'(c)={f(3)-f(0)}/{3-0}`.

Let us find such `c`.

The left-hand side is

`f'(c)=3c^2+1`.

The right-hand side is

`{f(3)-f(0)}/{3-0}={29-(-1)}/{3}=10`.

By setting them equal to each other,

#3c^2+1=10 Rightarrow 3x^2=9 Rightarrow x^2=3 Rightarrow x=pm sqrt{3}#

Since `0<c<3`, `c=sqrt{3}`.

I hope that this was helpful.