Tagged Questions

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Two Strictly Convex Functions with Contact of Order 1

Let $f,g: \mathbb{R}\rightarrow \mathbb{R}$ be two strictly convex functions, where $f$ is differentiable, $g$ is smooth, and $f\geq g$. Suppose that for some $x_0\in \mathbb{R}$: ...
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Differentiable Strictly Convex Function on Interval

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a differentiable, strictly convex function. Let $I\subset \mathbb{R}$ be a closed, bounded interval such that $f'(x) \neq 0$ on $I$. Is $f$ strongly ...
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Zero point when $f'(x)\gt c$

Suppose that the function $f:\mathbb R\to\mathbb R$ is continuously differentiable and that there is a positive number $c$ such that $f'(x)\ge c$ for all points $x$ in $\mathbb R$. Prove that there is ...
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Evaluate $\frac{\partial^2}{\partial t^2} \left[ \prod_{j=1}^k (1+t+\dots+t^{d_j -1}) \right]$ at $t=1$

I need to find a "nice" formula for the evaluation of $\frac{\partial^2}{\partial t^2} \left[ \prod_{j=1}^k (1+t+\dots+t^{d_j -1}) \right]$ at t=1, where $d_j \in \mathbb{N}$. I have already proved ...
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Assumptions in Word Problems.

My dilemma has been that I am confused on how we make mathematical assumptions in WORD problems. Suppose you are given a related-rates word problem. (Q#) Air is being pumped into a spherical balloon ...
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Does a nondecreasing, differentiable function have continuous derivative?

Are the following statements true? How to prove or disprove? (1). Let $f$ be a nondecreasing, differentiable function on $[0,1]$. Then $f$ is absolutely continuous? To be stronger, (2). Let $f$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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: ...
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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 ...
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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 $$f'(\xi)=\dfrac{f(\xi)-f(a)}{b-a}$$ ...
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$Df(x_0)$ 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 ...
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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. ...
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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 $0$. ...
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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 ...
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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 ...
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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 ...
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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 ...
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 ...