- #1
zeraoulia
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Homework Statement
The **mean value theorem** says that there exists a $c∈(u,v)$ such that $$f(v)-f(u)=f′(c)(v-u).$$ Here is my question.
Assume that $u$ is a root of $f$, hence we obtain $$f(v)=f′(c)(v-u);$$ assume that $f$ is a non-zero analytic function in the whole real line.
We can define the relation $ℜ$ by
$$cℜd⇔f′(c)=\frac{1-t}{1-u}f′(d)$$
where $u∈ℝ$ such that $f(u)=0$, $c∈(u,v)$ and $t∈ℝ$ such that $f(t)=0$, $d∈(t,v)$, that is we apply the mean value theorem for f in the two intervals $(u,v)$ and $(t,v)$, here $u,t,v$ are arbitrary.
Is $ℜ$ an equivalence relation? If so, determine its equivalence classes.
Homework Equations
The Attempt at a Solution
I have no idea to start with.