# Fréchet derivative and local maximum

I'm pretty confused with the idea of local maximum in function spaces. Normally having a null Fréchet derivative is a necessary but not sufficient condition for being a local maximum.

Computing the derivative

let $f:\mathbb{R} \mapsto \mathbb{R}$ be a continuous function. And lets denote the space of such functions $C_{\mathbb{R,R}}$.

$$F: C_{\mathbb{R,R}} \mapsto C_{\mathbb{R,R}}$$ $$F: f \mapsto \sin(f)$$

let us compute its derivative at point f:

$$D_F(f)h = \lim_{t\to 0} {F(f+th) - F(f) \over t}$$

where:

1. $f \in C_{\mathbb{R,R}}$
2. $g \in C_{\mathbb{R,R}}$
3. $t \in \mathbb{R}$

then:

$$D_F(f)h = \lim_{t\to 0} {\sin(f+th) - \sin(f) \over t}$$ $$D_F(f)h = \lim_{t\to 0} {\sin(f)\cos(th)+\cos(f)\sin(th) - \sin(f) \over t}$$ $$D_F(f)h = \lim_{t\to 0} {-h\sin(f) (1 - \cos(th))+h\cos(f)\sin(th) \over th}$$

So using: $$\lim_{x\to 0} {\sin(x)\over x} = 1$$ $$\lim_{x\to 0} {1-\cos(x)\over x} = 0$$

it reduces to:

$$D_F(f)h = h\cos(f)$$

Local maximum

we obviously have:

$$0 \leq ||F(f)||_\infty \leq 1$$

Hence:

if $||F(f)||_\infty = 1$, then f is a local maximum.

So any function $f$ such that $f(x) \equiv \pi \pmod \pi$ has a solution is a local maximum.

Null Fréchet derivative

$$D_F(f) = \cos(f) = 0 \Rightarrow \exists k \in \mathbb{N}, \forall x \in \mathbb{R}, f(x) = k\pi$$

such constant functions are indeed local maximums, but not the only ones. So instead of getting a superset containing all my local maximums, I get a strict subset of it from the nullity of my Fréchet derivative.

Question

As I'm pretty sure the mathematics I'm taught are right and I'm wrong... Where am I wrong ?

• What is the definition of the maximum of a mapping from $C(\mathbb R)$ to $C(\mathbb R)$? You seem to maximize $\|F(f)\|_\infty$ - the $\infty$-norm is not that differentiable. And $\cos(f)=0$ implies $\forall x\in \mathbb R$ $\exists k\in N$ such that $f(x)=k\pi$. Now find all such continuous functions... – daw May 15 '14 at 13:10
• @daw "the ∞-norm is not that differentiable", you mean there is a condition about the norm of my banach space for the fréchêt derivative to exists ? Or for a maximum do be defined ? – user2346536 May 15 '14 at 13:22
• @daw "Now find all such continuous functions" $\forall x f(x) = kπ$ seems a good shot to me. Constant functions basically. – user2346536 May 15 '14 at 13:32
• You seem to maximize $\phi(f):=\|F(f)\|_\infty$, but you only compute the derivative of $F$, but you do not check differentiability of $\phi$. – daw May 15 '14 at 13:38
• @daw You mean that, finding a maximum of a function $F:C_{\mathbb{R,R}} \mapsto C_{\mathbb{R,R}}$ does not really makes sens as $C_{\mathbb{R,R}}$ is not ordered. And so when I choosed my norm $||||_\infty$, I in fact was trying to maximize $||F(f)||_\infty$ ? Which is $\phi:C_{\mathbb{R,R}} \mapsto \mathbb{R}$ where R is order and $\phi(f) \leq \phi(f)$ does make sens. – user2346536 May 15 '14 at 13:44

The problem is to maximize $$\|F(f) \|_\infty.$$ However, the maximum is not differentiable. In order to perform the analysis, the function $f\mapsto\|F(f) \|_\infty$ needs to be differentiable