Derivatives & L'Hôpital's Rule Explained

[MrNotknowinganything]

Homework Statement

there be a function $f(x)$ continues in the section $[1,2]$ and Shearing in the Section $(1,2)$. Suppose also that takes place $\frac{f(2)}{f(1)}=2$. prove thet exists a $c \in(1,2)$ so that $f(c)=c \cdot f_{(c)}^{\prime}$
My best attempt :
\begin{array}{l}
1<c<2,1 \leq f(x) \leq 2 \\
f(c)=c f^{\prime}(c) \rightarrow c=\frac{f(c)}{f^{\prime}(c)} \\
f(2)=2 f(1)
\end{array}

Relevant Equations

there be a function $f(x)$ continues in the section $[1,2]$ and Shearing in the Section $(1,2)$. Suppose also that takes place $\frac{f(2)}{f(1)}=2$. prove thet exists a $c \in(1,2)$ so that $f(c)=c \cdot f_{(c)}^{\prime}$

Tried to use the information to put it in the definition of derivative and lopital but I couldn't get to anything

 

[PeroK]

I guess you have translated this question into English? I think I can work out that "shearing" means "differentiable".

According to the Homework guidelines you must post your best attempt to doing this yourself.

 

[MrNotknowinganything]

Tried to use the information to put it in the definition of derivative and lopital but I couldn't get to anything. This is my "attempt "

 

[PeroK]

Can you think of any theorems you have learned recently?

 

[MrNotknowinganything]

Can you think of any theorems you have learned recently?

Intermediate value test

 

[PeroK]

Intermediate value test

Any others? Like one that involves a derivative?

 

[MrNotknowinganything]

Any others? Like one that involves a derivative?

Derivative definition

 

[PeroK]

Derivative definition

That's not a theorem.

 

[MrNotknowinganything]

Derivative definition

Any others? Like one that involves a derivative?

Lopital rule

 

[PeroK]

That's for limits. What about the Mean Value Theorem?

 

[MrNotknowinganything]

That's for limits. What about the Mean Value Theorem?

I don't recall maybe I forgot , I'll be sure to check up on it now

 

[MrNotknowinganything]

That's for limits. What about the Mean Value Theorem?

Solved it thank you