: Higher derivatives<a class="btn btn-sm" href="/UUID/EDB/2D1/?lang=eng"><span class="ttfamily"><small class="scriptsize">[2D1]</small></span></a>

17.1 Higher derivatives[2D1]

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[1DD]Let $I$ be an open interval and $x_ 0∈ I$, let $f:I→ℝ$ be differentiable in $I$ and such that there exists the second derivative $f''$ in $x_ 0$: then show that the limit exists

\[ \lim _{t→ 0}\frac{f(x_ 0+t)+f(x_ 0-t)-2f(x_ 0)}{t^ 2} \]

and that it coincides with $f''(x_ 0)$.

Find then a simple example of $f$ differentiable in $(-1,1)$ and such that the second derivative $f''$ in $x_ 0=0$ does not exist, but the previous limit exists.

Hidden solution: [UNACCESSIBLE UUID ’1DF’]

E378

[1DG]5 Let $n≥ 1$ be an integer. Let $I$ be an open interval and $x_ 0∈ I$, let $f,g:I→ℝ$ be functions $n-1$ times differentiable in the interval, and whose $(n-1)$-th derivative is differentiable in $x_ 0$.

Show that the product $fg$ is differentiable $n-1$ times in the interval, and its $(n-1)$-th derivative is differentiable in $x_ 0$. Write an explicit formula for the n-th derivative $(fg)^{(n)}$ in $x_ 0$ of the product of the two functions, (formula that uses derivatives of only $f$ and only $g$).

(If you don’t find it, look in Wikipedia at the General Leibniz rule [ 55 ] ) .

Hidden solution: [UNACCESSIBLE UUID ’1DH’]

E378

[1DJ] Difficulty:*.Let $n≥ 1$ be an integer. Let $I,J$ be open intervals with $x_ 0∈ I,y_ 0∈ J$. Let then be given $g:I→ℝ$ and $f:J→ℝ$ such that $g(I)⊆ J$, $f,g$ are $n-1$ times differentiable in their respective intervals, their $(n-1)$-th derivative is differentiable in $x_ 0$ (resp. $y_ 0$) and finally $g(x_ 0)=y_ 0$.

Show that the composite function $f◦ g$ is differentiable $n-1$ times in the interval and its derivative $(n-1)$-th is differentiable in $x_ 0$.

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Then write an explicit formula for the nth derivative $(f◦ g)^{(n)}$ in $x_ 0$ of the composition of the two functions, (formula that uses derivatives of $f$ and $g$).
(If you can’t find it, read the wikipedia page [ 54 ] ; or, see this presentation: https://drive.google.com/drive/folders/1746bdJ89ZywciaEqvIMlGZ7kKHWVekhb ).

Hidden solution: [UNACCESSIBLE UUID ’1DK’]

E378

[1DM]Prerequisites:3,1.Show that the function

\begin{equation} 𝜑(x) = \begin{cases} e^{-1/x} & \text{if}~ ~ x>0 \\ 0 & \text{if}~ ~ x≤ 0 \end{cases} \label{eq:Cinfty_ non_ analitica} \end{equation}

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is of class $C^∞$, and for $x{\gt}0$

\begin{align*} 𝜑^{(n)}(x)= e^{-1/x} ∑_{m=1}^ n \binom {n-1}{m-1} \frac{n! }{ m! } \frac{(-1)^{m+n}}{x^{m+n}}\quad , \\ \quad \binom {n-1}{m-1}= \frac{(n-1)!}{(n-m)!(m-1)!}\quad . \end{align*}

whereas $𝜑^{(n)}(x)=0$ for each $n∈ℕ,x≤ 0$.

Proceed similarly to

\begin{equation} 𝜓(x) = \begin{cases} e^{-1/|x|} & \text{if}~ ~ ~ x≠ 0 \\ 0 & \text{if}~ ~ x= 0 \end{cases} \label{eq:Cinfty_ non_ analitica_ bis} \end{equation}

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again $𝜓∈ C^∞$ and $𝜓^{(n)}(0)=0$ for each $n∈ℕ$; but in this case $𝜓(x)=0\iff x=0$. Hidden solution: [UNACCESSIBLE UUID ’1DN’][UNACCESSIBLE UUID ’1DP’][UNACCESSIBLE UUID ’1DQ’]

E378

[1DR] Let it be given $N$ positive integer. Find an example of a function $C^∞$ with $𝜑(x)=0$ for $x{\lt}0$ while $𝜑^{(n)}(x){\gt}0$ for $0≤ n≤ N$ and $x{\gt}0$.

Hidden solution: [UNACCESSIBLE UUID ’1DS’] Note however that it cannot be required that all derivatives be positive, because of exercise 2.

E378

[1DT] What can you put in place of "???" so that the function

\[ g(x) = \begin{cases} ??? & \text{if}~ ~ ~ 0{\lt}x{\lt}1~ , \\ {} 1 & \text{if}~ ~ x≥ 1 ~ ,\\ {} 0 & \text{if}~ ~ x≤ 0~ . \end{cases} \]

is $C^∞$?

More generally, how can two $C^∞$ functions be connected, so that the whole function is $C^∞$? Given $f_ 0,f_ 1∈ C^∞$, show ¹ that there is a function $f∈ C^∞$ that satisfies

\begin{eqnarray*} f(x) = f_ 0(x)& \text{if}~ ~ ~ x≤ 0 \quad ,\\ {} f(x) = f_ 1(x)& \text{if}~ ~ ~ x≥ 1 \quad . \end{eqnarray*}

Hidden solution: [UNACCESSIBLE UUID ’1DV’]

E378

[1DW]Difficulty:*.Let $f_ 0,f_ 1:ℝ→ℝ$, $f_ 0,f_ 1∈ C^∞$ with $f'_ 0,f'_ 1{\gt}0$ and $f_ 1(1){\gt}f_ 0(0)$: then one can interpolate with a function $f∈ C^∞$ that satisfies

\begin{eqnarray*} f(x) = f_ 0(x)~ ~ \text{if}~ ~ ~ x≤ 0 \\ f(x) = f_ 1(x)~ ~ \text{if}~ ~ ~ x≥ 1 \end{eqnarray*}

so that the interpolant has $f'{\gt}0$.

What if $f_ 1(1)=f_ 0(0)$?

Hidden solution: [UNACCESSIBLE UUID ’1DX’]

E378

[1DZ]Prerequisites:4. Find an example of function $f:ℝ→ℝ$ with $f∈ C^∞$ and such that, setting $A=\{ x:f(x)=0\} $, the point $0$ will be the only point of accumulation of $A$, i.e. $D(A)=\{ 0\} $. Compare this example with Prop. 6.8.4 in the notes [ 2 ] ; and with the example 2. Hidden solution: [UNACCESSIBLE UUID ’1F0’]

E378

[1F1] Difficulty:*.Note:Hadamard’s lemma.

Let $f:ℝ→ℝ$ be a function of class $C^∞$, and such that $f(0)=0$. Define, for $x≠ 0$, $g(x){\stackrel{.}{=}}f(x)/x$. Show that $g$ can be prolonged, assigning an appropriate value to $g(0)$, and that the prolonged function is $C^∞$. What is the relationship between $g^{(n)}(0)$ and $f^{(n+1)}(0)$?

Hidden solution: [UNACCESSIBLE UUID ’1F2’]

E378

[1F4]Prerequisites:9.Difficulty:*.Let $f:ℝ→[0,∞)$ be a function of class $C^∞$ such that $f(0)=0$, $f(x){\gt}0$ for $x≠ 0$, and $f''(0)≠ 0$: show that

\[ g(x)= \begin{cases} \sqrt{f(x)} & se ~ x≥ 0\\ -\sqrt{f(x)} & se ~ x{\lt} 0 \end{cases} \]

is of class $C^∞$. Hidden solution: [UNACCESSIBLE UUID ’1F5’]

E378

[1F7]Difficulty:* . Given $x_ 0{\lt}x_ 1{\lt}x_ 2{\lt}\ldots {\lt}x_ n$ and given real numbers $a_{i,h}$ (with $i,h=0,\ldots n$) show that there is a polynomial $p(x)$ such that $p^{(i)}(x_ h)=a_{i,h}$.

This result is the starting point of the Hermit method of polynomial interpolation, see [ 57 ] .

Hidden solution: [UNACCESSIBLE UUID ’1F8’]

E378

[1F9] Prerequisites:convex functions.Note:Exercise 1, written exam March 1st, 2010.

Let’s consider the functions $f:ℝ→ℝ$ of class $C^∞$, such that for every fixed $n≥ 0$, $f^{(n)}(x)$ has constant sign (i.e. it is never zero) ² . We associate to each such function the sequence of signs that are assumed by $f,f',f''\ldots $.

What are the possible sequences of signs, and what are the impossible sequences?

(E.g. for $f(x)=e^ x$, the associated sequence is $+++++\ldots $, which is therefore a possible sequence.)