Inverse Theorem and Fermat's Theorem (Iso point)

[Enzipino]

Hello,
I've been using Caratheodory's Lemma to prove the Inverse Function Theorem and Fermat's Theorem. I have managed to prove both of them, I would just like someone to look over my proof and tell me if I'm missing anything (i.e. should I clarify any parts of my proof). So here goes:

Inverse Function Theorem - Suppose that $f$ is a one-one continuous function (this means it's bijective right?) on an interval $S$, $f$ is differentiable at a given number ${x}_{0}\in S$, and $f'({x}_{0})\ne0$. Then $f^{-1}$ is differentiable at the number $f({x}_{0}) = {y}_{0}$ and we have $(f^{-1})'({y}_{0})=\frac{1}{f'({x}_{0})}$

Proof: Since $f$ is differentiable at $x_0$, $\exists\phi:S\to\Bbb{R}$ such that $\phi(x_0)=w$ and $f(x) = f(x_0) + (x-x_0)\phi(x)$ [Caratheodory's]
Let $x=f^{-1}(y)$, we have $f(f^{-1}(y)) = f(f^{-1}(y_0)) + (f^{-1}(y)-f^{-1}(y_0))\phi(f^{-1}(y))$.
$\implies f(f^{-1}(y))-f(f^{-1}(y_0)) = (f^{-1}(y)-f^{-1}(y_0))\phi(f^{-1}(y))$
$\implies y-y_0=(f^{-1}(y)-f^{-1}(y_0))\phi(f^{-1}(y))$
$\implies \frac{y-y_0}{(f^{-1}(y)-f^{-1}(y_0))}=\phi(f^{-1}(y))$
Flipping both sides of the equation we have $\frac{(f^{-1}(y)-f^{-1}(y_0))}{y-y_0}=\frac{1}{\phi(f^{-1}(y))}$
$\implies (f^{-1})'({y}_{0})=\frac{1}{\phi(x)}$
$\therefore(f^{-1})'({y}_{0})=\frac{1}{f'(x_0)}$
Q.E.D.​

For Fermat's Theorem, my book has it in 5 parts but I'm just going to post the 5th part which is the main theorem. I will be using 2 other parts concerning the maximum:
1. If $x$ is not the right endpoint of $S$ and $f'(x)>0$, then $f$ cannot have a max at $x$.
2. If $x$ is not the left endpoint of $S$ and $f'(x)<0$, then $f$ cannot have a max at $x$.

Fermat's Theorem - Suppose that $f$ is a function defined on an interval $S$ and that $f$ is differentiable at a number $x\in S$. If $x$ is not an endpoint of $S$ and $f$ has either a maximum or minimum value at $x$, then we must have $f'(x)=0$.

Proof: Suppose that $f$ has a maximum value at $x$. Using the converse of 1 and 2 above, we have that $f'(x)\le0$ and $f'(x)\ge0$. Thus, $f'(x)=0$.
Q.E.D.​

This proof just seems way too easy which has me second guessing myself but I know that the theorem definitely follows from the other parts so by proving those parts, it should lead me to proving this final part.

Any feedback is greatly appreciated. Any tips about how to explain these two things in front of an entire class would be greatly appreciated as well. Thanks!

 

[jvicens]

Thank you for sharing your proof of the Inverse Function Theorem and Fermat's Theorem. Overall, your proof looks sound and well-organized. However, I have a few suggestions for clarification and further explanation.

Firstly, you are correct in stating that a one-one function is also bijective. However, it may be helpful to explicitly state this in your proof, especially if you are presenting it to a class. Additionally, you may want to define what it means for a function to be "continuous on an interval," as this is a key assumption in the theorem.

Secondly, when using Caratheodory's Lemma, it may be helpful to explain why this lemma is necessary for the proof. This will help your audience understand the motivation behind the steps you are taking.

In your proof of the Inverse Function Theorem, you state that $\phi(x_0)=w$ and $f(x) = f(x_0) + (x-x_0)\phi(x)$. It would be helpful to explain why this is the case and how it relates to the differentiability of $f$ at $x_0$. Additionally, you may want to clarify why you chose to use $y$ instead of $y_0$ in the following steps: $y=f(x)$ and $y_0=f(x_0)$. This may be confusing for some readers.

For your proof of Fermat's Theorem, it would be helpful to explain why the converse of parts 1 and 2 leads to $f'(x)=0$. This may not be immediately obvious to some readers. Additionally, you may want to clarify what is meant by "maximum or minimum value at $x$." This could refer to a local or global maximum/minimum, and it may be helpful to specify which one you are considering in this proof.

Overall, your proofs are clear and well-structured. However, providing more explanation and clarification will make them more accessible to a wider audience. Best of luck with your presentation!