-1
votes
2answers
49 views

Differentiability at x=0 [closed]

Discuss the differentiability of the following function in $x$ = $0$: $ f:\mathbb{R} \to \mathbb{R}: x\mapsto \begin{equation} f(x)= \begin{cases} \sqrt{x} & \text{if } x \geq 0 \\-\sqrt{-x} & ...
1
vote
1answer
34 views

Having derivative in some $x_{0}$ implies having it in $U(x_{0})$ [closed]

Let $f$ be continious function in R and it has a derivative in $x_{0}$. Does it have derivative in some $U(x_{0})$?
2
votes
3answers
178 views

Derivative and integral of the abs function

I would like to ask about how to find the derivative of the absolute value function for example : $\dfrac{d}{dx}|x-3|$ My try:$$ f(x)=|x-3|\\ f(x) = \begin{cases} x-3, & \text{if }x \geq3 \\ ...
8
votes
3answers
71 views

Show that it is possible that the limit $\displaystyle{\lim_{x \rightarrow +\infty} f'(x)} $ does not exist.

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ a differentiable function with continuous derivative and the limit $\displaystyle{\lim_{x \rightarrow +\infty} f(x) }$ exists. Show with an example that it ...
2
votes
1answer
67 views

Characterization of differentiable functions from $\mathbb{R}^m$ to $\mathbb{R}^n$.

Let $U\subset\mathbb{R}^m$ be an open set. Consider a function $f:U\to\mathbb{R}^n$ and a point $a\in U$. I need help to prove that the following sentences are equivalents. (a) There exists a ...
0
votes
1answer
26 views

relations between differential, partial derivative, directional derivative

I am a bit lost. Could you explain me relations between differential, partial derivative, directional derivative? I mean that I need some theorem and proofs that for example if differential exists ...
2
votes
2answers
28 views

mean value property of derivatives in high dimensions

Let $E$ be a path-connected subset of $\mathbb{R}^n$ and $f$ a differentiable function on $E$. Prove or disprove: for any $x,y\in E$, there exists $z\in E$ such that $f(x)-f(y)=\nabla f(z)\cdot ...
0
votes
1answer
22 views

Fractional Derivatives on a function with bounded Support

I have a question about functions that have bounded support in $\mathbb{R}$. In particular, suppose that I have a function $f$ with support $A\subset \mathbb{R}$ so that $A$ is compact. Without loss ...
1
vote
2answers
47 views

Extreme value problem, maximize ratio of volume to surface area

For a cylindrical can, how to choose the ratio of the height to the radius such that the ratio of the volume to the surface area gets maximized? The volume ist $V = \pi r^2 h$ and the surface ...
2
votes
1answer
61 views

A calculus problem

Question: Suppose that $u(x,t)$ is continuous, together with its first and second partial derivatives; suppose that $u$ and its first partial derivatives are periodic in $x$ of period $1,$ and ...
6
votes
2answers
164 views

simple way to show $|| \partial_x \int_{B(x,\epsilon)} \frac{x-y}{|x-y|^3} f(y) dy||_{\infty} = O(||f||_{\infty})$ in $\mathbb{R}^3$

We are set in $\mathbb{R}^3$. Let $f: \mathbb{R}^3 \rightarrow \mathbb{R}$ be a $C^1_0$ function, i.e. continuously differentiable with compact support. Let $\epsilon > 0$ be small. I need to show ...
2
votes
1answer
41 views

Motivation and Derivation of the Riccati Equation Transformation

Given a Riccati Equation which is differential equation of the form: $$ \frac{dy}{dx} = a_0 (x) + a_1 (x)y + a_2 (x)y^2 $$ It is well known that the transformation: $$ y = -\frac{1}{a_2(x)} ...
0
votes
1answer
43 views

Construct a continuous function which has no derivative almost everywhere.

Georg Cantor is famous for the first set theory (in "naive" terms) and the diagonal argument. However Cantor is also credited with the Cantor Set and for constructing a continuous function which has ...
0
votes
0answers
20 views

Operator differentiability

I was wondering, what techniques can one use to prove that an operator (let's say acting on real analytic functions and taking values in a Banach space) is infinitely differentiable? I know that, for ...
2
votes
1answer
18 views

partial derivative of function with a matrix

Let $A$ be a $n\times n$ matrix. Let $f\in C^1(\mathbb R^n)$ and $g:\mathbb R^n\rightarrow\mathbb R, g(x)=f(Ax)$. What is the partial derivative $\partial_{x_i} g(x)$? So $Ax=(\sum_{l=1}^n ...
1
vote
1answer
56 views

Write this ODE without any square roots

Given the function $$u(t):=\sqrt{\sum_{i=0}^n \alpha_i t^{2i}}$$ is it possible to plug this into the ODE $$(t^2-1)u''(t)+tu'(t)(1-8a+8at^2)-4(a+a^2-2at^2+n(-a+2at^2)-C)u(t)=0 $$ such that I get a ...
0
votes
1answer
44 views

Show that function is partially differentiable

I have the following function: $$F: \mathbb{R}^2 \rightarrow \mathbb{R}, ~~ (x,y) \rightarrow xy\frac{x^2-y^2}{x^2+y^2}$$ for $(x,y) \ne 0$ and $F(0,0) = 0$. I want to show that $F$ is partially ...
4
votes
1answer
38 views

Differentiable Path of Operators and their Inverses

Let $\mathcal{H}$ be a separable Hilbert space. Consider a differentiable map $\mathbb{R} \rightarrow \mathcal{B}(\mathcal{H}), t \mapsto A(t)$, where $\mathcal{B}(\mathcal{H})$ is the space of ...
1
vote
1answer
77 views

Why continuity at a point one of Dini derivatives implies the continuity at this point others Dini derivatives?

Let $f:[a,b] \rightarrow \mathbb R$ be a continuous function and $x_0\in (a,b)$. How to prove that if the Dini derivatives $D^+f(x_0)$ is finite and continuous at $x_0$ then also $D_+f(x_0)$ is ...
5
votes
1answer
124 views

How to show that $e^x$ is differentiable?

I tried to search for a few minutes but I didn't find this question so I hope it's not a duplicate. So I want to show that $(e^x)' = e^x$. To do that, I must proof that the limit: ...
9
votes
1answer
245 views

Find compressed form for cumbersome calculation

Given the three functions $u^{\mathrm{(I)}}(t)\;=t \left(t^2\right)^{k}\,e^{2\beta t^2},\\ u^{\mathrm{(II)}}(t)=\sqrt{1-t^2}\left(t^2\right)^{k}\,e^{2\beta t^2},\\ ...
0
votes
0answers
14 views

Maximum Principle - Proof

We want to show the maximum principle for a function $f = f(x,t)$ on a n-dimensional hypersurface $M,$ that is, (Corollary) Let $f = f(X,t)$ be a function on M, let $\vec{a}$ be a vector field on ...
1
vote
1answer
38 views

Question about limits and Mean Value Theorem

Let $f:(a,b) \rightarrow \mathbb{R}$ and $g:(a,b) \rightarrow \mathbb{R}$ be differentiable on (a,b) with $g'(x) \neq 0$ for all $x$ in $(a,b)$. Suppose $\lim_{x \to b-}\dfrac{f'(x)}{g'(x)}$ ...
1
vote
1answer
39 views

Dini Derivative

Let $f$ be defined on $\mathbb{R}$ such that $$ f(x) = \begin{cases} |x|, & \text{if }x \in \mathbb{Q} \\ |2x|, & \text{if }x \notin \mathbb{Q} \end{cases} $$ Calculate ...
0
votes
0answers
26 views

Gradient; how to do this?

I want to do this gradient, but I just don't get the right result: $\phi: \mathbb{R}^3 \rightarrow \mathbb{R}$ and $F(Y) = - q \ \text{grad}\phi(Y) = \frac{1}{4 \pi \varepsilon_0} ...
2
votes
1answer
37 views

right derivative of a continuous function

Let $f:(a,b)\longrightarrow \mathbb{R}$ be continuous. Suppose $D_+f(x)=\lim_{h\to 0+}\frac{f(x+h)-f(x)}{h}\geq 0$ for any $x\in (a,b)$. Prove that $f(x_1)\geq f(x_0)$ whenever $x_1\geq x_0$. How to ...
13
votes
2answers
212 views

How prove that there exists $\xi\in(a,b)$ with $f'(\xi)=\frac{f(\xi)-f(a)}{b-a}$

Let $f(x)$ be continuous on $[a,b]$, differentiable on $(a,b)$, and with some $c\in(a,b)$ such that $f'(c)=0$. Show: There exists $\xi\in(a,b)$ such that $$ f'(\xi)=\dfrac{f(\xi)-f(a)}{b-a} $$ ...
0
votes
1answer
30 views

$Df(x_0)$ is one-to-one. show $f$ is one-to-one on a neighborhood of $x_0$

Suppose $f:\mathbb{R}^n \to \mathbb{R}^m$ is $C^1$ and $Df(x_0)$ is one-to-one. Show $f$ is one-to-one on a neighborhood of $x_0$. I think it's about inverse function theorem. but i cannot prove ...
1
vote
0answers
38 views

Meaning of this differentiation operators

I have been just reading this paper here: paper and was wondering how they carry out the differentiation in (4.9). In principle, this should be just the differentiation of 4.8 with the help of 4.7a. ...
1
vote
0answers
22 views

Show that both these Prove $f$ is differentiable from the right at $0$

Let $f:[0,1) \rightarrow \mathbb{R}$ be continuous on [0,1) and differentiable on $(0,1)$. Suppose the limit $\lim_{x \to +0} f'(x)$ exists. Prove that $f$ is differentiable from the right at $0$. ...
1
vote
1answer
71 views

How to show that this function is differentiable?

Let $$\phi: \mathbb{R} \rightarrow \mathbb{\mathbb{C}}, s \mapsto \int_2^{\infty} \frac{e^{isx}}{x^2\ln(x)}dx$$, I want to show that this function is differentiable everywhere. Unfortunately, it ...
0
votes
0answers
30 views

Is it correct that $\frac{d\theta}{d\varphi'}=\left(\frac{d\varphi'}{d\theta}\right)^{-1}$?

Let $\theta$ be $k\times1$ and $\varphi$ be $k\times1$. Then $\frac{d\theta}{d\varphi'}$ is a $k\times k$ matrix with entries $\frac{d\theta_{i}}{d\varphi_{j}},\,\, i,j=1,...,k$. I was wondering if ...
3
votes
0answers
35 views

Prove the following expression is true.

Let $x_1,...,x_{n+1}$ be arbitrary points in $[a,b]$ and let $$Q(x)= \prod\limits_{i=1}^{n+1} (x-x_i)$$Now suppose $f$ is an n times differentiable function and tha P is a polynomial function of ...
0
votes
1answer
11 views

Functions with symmetrical behaviour with respect to an axis or a plane

Suppose we have two functions with a symmetrical behaviour with respect to an axis. For the sake of simplicity, let $f(x)$ and $g(x)$ have a symmetrical behaviour with respect to the $y$ axis. A ...
1
vote
1answer
12 views

Proof about First order derivative

Show that if $f'(c)>0$ then there exists $\delta>0$ such that $x \in (c,c+\delta) \ \ \implies \ \ f(x)>f(c)$ $x \in (c-\delta,c) \ \ \implies \ \ f(x)<f(c)$ My Attempt Now ...
4
votes
1answer
134 views

Showing $f^{(n-1)}(\xi) = 0$ for some $\xi$

Let $f$ be an $n$ times differentiable function on the interval $A$. If $x_1 < x_2 < \cdots < x_p$ are points on $A$ and $n_i, 1 \leq i \leq p,$ are natural numbers such that $n_1 + n_2 + ...
1
vote
1answer
23 views

For what interval does this power series converge and for what interval does it determine a differentiable function?

For what range of values of $x$ does $\sum_{n=1}^{\infty } \dfrac{1}{n}(1+\sin x)^n$ converge? Find with proof an interval on which it determines a differentiable function of $x$ and show that ...
0
votes
0answers
32 views

Suppose $f$ is twice differentiable, prove;

Attempt; First Part Here I just have to stat the definition of Mean Value Theorem; If $g$ is continuous on $[z-a,z+a]$ and differentiable on $(z-a,z+a)$ the there exists $\xi \in (z-a,z+a)$ such ...
1
vote
1answer
29 views

One sided limits equal to actual limit

Suppose $f:(a,b) \backslash \{c\} \rightarrow \mathbb{R}$ is a function such that $$\lim_{x \to \ c+}f(x) \ \ \ \ and \ \ \ \ \lim_{x \to \ c-}f(x)$$both exists and are equal to a common value ...
2
votes
1answer
53 views

Difficulty in proving this inequality

Let $f \in C^{(n)}(-1,1)$ and $\sup_{-1 <x< 1}|f(x)|\leq 1$. Let $m_k(I) = \inf_{x \in I} |f^{(k)}(x)|$, where $I$ is an interval contained in $(-1,1)$. If $I$ is partitioned into three ...
0
votes
2answers
45 views

Let $f:\mathbb R \rightarrow \mathbb R$ continuous function such that $f(x+y)=f(x)+f(y)$. Then there exist a real $r$ such that $f(x) = r x$.

Let $f:\mathbb R \rightarrow \mathbb R$ continuous function such that $f(x+y)=f(x)+f(y)$. Then there exist a real $r$ such that $f(x) = r x$. My try: with $f(\frac1q x)$: $$f(x) = f(q\cdot\frac1q ...
3
votes
2answers
212 views

Two halls 6 and 9 meters perpendicularly intersect. Optimization

Two halls 6 and 9 meters perpendicularly intersect. Find the length of the longest straight bar to be passed horizontally from one aisle to another by a corner without deformation. and this is my ...
0
votes
0answers
22 views

Question about a proof, monotone differentiation theorem

I have some problems understanding some technical details in the proof of monotone differentiation theorem in Donald L. Cohns book on measure theory (Theorem 6.3.3). Let $F: R \rightarrow R$ be ...
0
votes
2answers
42 views

Derivatives and what is a good definition?

So I have a question in general about derivatives. I understand that the formal definition is something like $f$ is differentiable at $x=a$ if the limit exists where that limit is either the limit as ...
2
votes
0answers
56 views

how to show ${\partial ^2 f \over \partial x \, \partial y}= {\partial ^2 f \over \partial y\, \partial x}$ ??

$f(x,y)$ is real-valued function on $R^2$ $f$ is of class $C^1$ and $\dfrac{\partial ^2 f}{\partial x \,\partial y}$ exists and is continous. how to show $\displaystyle{\partial ^2 f \over \partial ...
0
votes
1answer
45 views

Show $f$ differentiable $\iff$ $f(x) -f(x_0) = \psi(x)(x-x_0)$

The questions are a) to show that $f(x)$ is differentiable at the point $x_0$ if and only if $f(x) -f(x_0) = \psi(x)(x-x_0),$ where $\psi(x)$ is a function that is contiuous at $x_0$ b) if ...
0
votes
1answer
41 views

For which $\alpha$ does this function have a positive solution?

For which $\alpha \in \mathbb{R}$ does $$e^{\alpha x}-1=x$$ have a positive solution Hint; Consider Derivatives at $0$. My attempt; Now firstly I will rewrite this to form the function ...
0
votes
1answer
19 views

Proving a function is bounded above.

Hi all, while doing this question ,I feel that I understand the concept of the question, but can't seem to formulate it into a viable answer. If the limit as $x \rightarrow \infty$ is the same as $x ...
6
votes
4answers
405 views

Construct a function that is nowhere differentiable.

I have been working on this question for a very long time now and seem to have reached a dead end, I will show all my attempted solutions, and any help on the various parts of the question would be ...
1
vote
1answer
41 views

find the maximum of the function F under the condition $ \sum_{i=1}^N x_i = 1$

Let F a function of $ \mathbb{R} ^N_+ \rightarrow \mathbb{R}$ defined as : $$F(x_1,..,x_N)= - \sum_{i=1}^N x_i log(x_i) , x_i \gt 0$$ How can i find the maximum of the function F under the ...