0
votes
0answers
25 views

Proving concavity for complicated function

I have a rather complicated function, $f$, that I am trying to demonstrate is log-log-concave, i.e., $$\frac{d^2\log f}{(d\log x)^2}\leq 0.$$ The reason I think it is concave is purely heuristic. ...
1
vote
2answers
81 views

Assumptions that can be made for $f(x) + xf '(x)\leq 0$

I am wondering if we can make any assumptions about a function $f$ i.f.f. it satisfies $$f(x) + xf '(x)\leq 0 \qquad\forall \;x>0\;?$$
0
votes
2answers
44 views

Higher-order derivative test

Let $f:I\rightarrow \Bbb{R}$ $2007$ times differentiable at $x_0 \in I$. Also: $f'(x_0) = f''(x_0) = ... = f^{(2006)} = 0$ but $f^{(2007)} > 0$. Prove there's $\delta> 0$ such that $f$ is ...
1
vote
1answer
28 views

Find the largest $n\in \Bbb{N}$ answering the following terms

Let $$f(x) = -\frac{1}{12}x^4 + o(x^5)$$ Also, Let $$g(x) = \begin{cases} \frac{f(x)}{x^n} &\mbox{if } x\ne 0 \\ C & \mbox{if } x=0 \end{cases}$$ I need to find the largest $n\in\Bbb{N}$ ...
1
vote
1answer
27 views

Multivariable-calculus, derivative and second derivative [closed]

I got the function $f(x,y)=\ln \sqrt{x^2+y^2}$. The task is to find the derivative function and the second derivative function. How do I get there?
2
votes
3answers
63 views

show that $f^{(3)}(c) \ge 3$ for $c\in(-1,1)$

Let $f:I\rightarrow \Bbb{R}$, differetiable three times on the open interval $I$ which contains $[-1,1]$. Also: $f(0) = f(-1) = f'(0) = 0$ and $f(1)=1$. Show that there's a point $c \in (-1, 1)$ ...
1
vote
3answers
41 views

The inflection points of $f(x)=(x^2-4x+1)e^{-x}$

I got the function $f(x)=(x^2-4x+1)e^{-x}$. The task is to find the inflection points. The correct answer is $x=4-\sqrt{5}$ and $ x=4+\sqrt{5} $. I got the second derivative to $f(x)$. But when I ...
1
vote
1answer
40 views

Differentiable functions and examples

can someone give me an example of Differentiable function at x=4 and funcstions who dont Differentiable function at x=4? $f(x) = 2x-7$ $k(x) = 100x^7-55x^5+10000x^2$ $g(x) = 23$ Those are ...
6
votes
2answers
82 views

$f$ is twice differentiable, $f + 2 f^{'} + f^{''} \geq 0$ , prove the following

Let $ f : [0,1] \rightarrow R$. $f$ is twice diff. and $f(0) = f(1) = 0$ If $f + 2 f^{'} + f^{''} \ge 0$ , prove that $f\le 0$ in the domain. Don't give complete solution, only hints.
3
votes
0answers
38 views

Find all differentialbles function [closed]

Find all differentialbles function $f:[0,\infty)\rightarrow\mathbb R$ such that: a) $f^{\prime}$ is non-decreasing; b) $x^{2}f^{\prime}(x)=f^{2}(f(x)),~\forall x\in\lbrack0,\infty)$
8
votes
3answers
212 views

Determining the maximum value for the solution of this delay differential equation?

I am working on the following delay differential equation $$\frac{df}{dt}=f-f^3-\alpha f(t-\delta)\tag{1},$$ where $\frac{1}{2}\leq\alpha\leq 1$ and $\delta\geq 1$. I know that there are three ...
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 $U(x_{0})$?
0
votes
4answers
69 views

Derivative with respect of a function

i have a function of two variables: $f(\theta,\phi) = \theta \sin(\phi)$ and i have to differentiate $f(\theta,\phi)$ with respect to: $1 - 0.5\theta^2$ That is: ...
0
votes
2answers
72 views

Gradients and functions on matrices

Given a twice differentiable $f: \Bbb R \to \Bbb R$, with continuous second order derivative. We define $$F(x) = \sum_{i=1}^{m}f(x_i)$$ and $$L(x) = \sum_{i=1}^{m}f( \langle a_i, x \rangle+ b_i),$$ ...
1
vote
1answer
35 views

Evaluate position of first secondary maximum of $\frac{\sin N (x/2)}{\sin (x/2)}$

The function $$f(x) = \displaystyle \left | \frac{\sin \left( N \displaystyle \frac{x}{2} \right)}{\sin \left( \displaystyle \frac{x}{2} \right)} \right |$$ when evaluated for $x > 0$, has its ...
2
votes
1answer
58 views

Figuring out when $f(x) = \sin(x^2)$ is increasing and decreasing

Regarding the function $f(x) = \sin(x^2)$, I'm supposed to figure out when it is increasing/decreasing. So far, I've found the derivative to be $f'(x) = 2x\cos(x^2)$. So long as I can solve the ...
2
votes
2answers
114 views

$f\left(\frac{1}n\right)=\frac{n^2}{n^2+1}$

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ of class $C^\infty$ $\forall n\in\mathbb{N}^*,f\left(\frac{1}n\right)=\frac{n^2}{n^2+1}$ Let $p\in\mathbb{N}^*$ What is the value of $f^{(p)}(0)$ ? (by ...
11
votes
1answer
105 views

Differentiation of a function $f:\mathbb{Q}\to \mathbb{Q}$(Rational Calculus)

Assume that $f:\mathbb{Q}\to \mathbb{Q}$ is given such that $\forall a\in \mathbb{Q}$ the following limit, exists \begin{equation} \lim_{x\to a} \frac{f(x)-f(a)}{x-a}\in \mathbb{R} ...
0
votes
1answer
40 views

Finding Derivative of a complicated function

I have the following function I want to be able to feed into the Newton-Rhapson root solving algorithm (C++ boost), since an analytical solution is not possible:$$f(p, k, n, Pr)=\left(\sum_{i = ...
1
vote
1answer
83 views

Prove there's $x_0$ such that $f'(x_0)=0$

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ differentiable at $\mathbb{R}$ and: $$\lim_{x\rightarrow \infty}\left( f(x)-f(-x) \right) = 0$$ Show there's $x_0$ such that $f'(x_0) = 0$. I tried to use ...
1
vote
1answer
54 views

Is there an actual function that “escapes” from zero after an amount of derivatives?

Suppose we have a function $f$ such that for all $i=2,...k: f^{(i)}=0$ but for $i\ge k+1$ we have $f^{(i)}\neq 0$. Can there be such functions in theory ? and is there an actual function that ...
1
vote
0answers
43 views

$f'(x)$ to $f(x)$ is it possible without knowing the value of $f'(x)$ or $f(x)$

I don't know much math and i got stuck at a problem: I'm not sure if it possible how to do it. I must use hermite interpolation for the following: 'Find the polynomial interpolating the function $f$ ...
1
vote
1answer
40 views

Reciprocal Rule Question

$f(z) = \frac{4 z + 4}{z^2 + z + 1}$ I have a question about the reciprocal rule concerning derivatives. I want to work around doing the quotient rule for the above function if possible, but I am not ...
2
votes
1answer
160 views

Find fourth derivative of function

The function is $\displaystyle{\frac{3x^4}{1-x}}$ and I am trying to find $\displaystyle{\frac{d^4}{dx^4}}$. However, I want to find the solution without using the quotient rule $4$ times in a row. I ...
0
votes
1answer
77 views

I can't seem to find this derivative any help would be great.

A rocket of mass m = 1000 kg is traveling in a straight line for a short time. The distance in meters covered by the rocket during this time is described by the function $r(t)=t^3 −3t^2 +6t$ where ...
1
vote
1answer
22 views

If $g(x) = \text{arctanh}\ (\log x)$, find $g'(x)$.

If $g(x) = \text{arctanh}\ (\log x)$, find $g'(x)$. I tried to separate the terms first and I got $\dfrac12 (\log(1+\log x) - \log(1-\log x))$. The answer is $\dfrac1{x(1-\log x)^2}$.
6
votes
6answers
202 views

Prove that $f(x)\equiv0$ on $\left[0,1\right]$

Let $f(x)$ differentiable on $\left[0,1\right]$ such that $f(0) = 0$. Also, assuming that $\forall x\in \left[0,1\right]:\left|f'(x)\right| \le \left|f(x)\right|$. Prove that $f(x)\equiv 0$ What I ...
1
vote
2answers
28 views

A question on a multivariable continuously differentiable function

Assume $f(x_{1},x_{2})$ is a real-valued continuously differentiable function, and assume it holds that $x_2D_{1}f(x_1,x_2) - x_1D_2f(x_1,x_2) = 0$ where $D_1$ is the partial derivative with respect ...
2
votes
1answer
33 views

Can any function on naturals be interpolated to a smooth function on reals?

Let $f : \mathbb{N} \rightarrow \mathbb{N}$ be an arbitrary function from naturals to naturals. Is it always possible to find a function $g : \mathbb{R} \rightarrow \mathbb{R}$ such that for any $n ...
1
vote
1answer
53 views

Function which derivative at $0$ is $1$ but is not monotonic increasing

Please, I need help in order to understand the following assumption that I've found in Bartle's book Introduction to Real Analysis page 171. It says: One might suppose that,if the derivative is ...
1
vote
1answer
31 views

Finding the point where a function turns smaller then another

Sorry, couldn't explain better on the title. I mean, if you have a function for the income over time $I(t)$ and another one for costs $C(t)$ and you want to find out the time $t$ for which the profit ...
0
votes
1answer
26 views

Differential Calculus Problem - Sphere volume increasing (differentiation of algebraic functions)

The Air is pumped into a spherical ball which expands at a rate of 8cm^3 per second. Find the exact rate of increase of the radius of the ball when the radius is 2 cm. I have tried this question, ...
6
votes
1answer
264 views

Can monsters of real analysis be tamed in this way?

Consider the Weierstrass Function (somewhat generalized for arbitrary wavelengths $\,\lambda > 0$ ): $$ W(x) = \sum_{n=1}^\infty \frac{\sin\left(n^2\,2\pi/\lambda\,x\right)}{n^2} $$ $W(x)$ is an ...
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 ...
2
votes
1answer
64 views

Does the function differentiable?

Let $\alpha, \beta \ge 1 \in \mathbb{N}$ and: $$f(x) = \left\{ {\matrix{ {{x^\beta }\sin \left( {{1 \over {{x^\alpha }}}} \right)} \cr {0,x = 0} \cr } } \right.$$ I checked for ...
1
vote
0answers
12 views

unimodality and continuous

i would like to ask question about unimodality of probability function ,from wikipedia http://en.wikipedia.org/wiki/Unimodal it says that In mathematics, unimodality means possessing a unique mode. ...
4
votes
2answers
58 views

Are $C^{\infty}$ completely defined by their derivatives?

This question has been on my mind for some time. Here's my process. Firstly, is it possible to construct a function such that it's defined with a different expression on different intervals, but that ...
2
votes
1answer
48 views

Let $z=\ln \tan\frac xy.$ What is $z_x$ and what is $z_y$?

Let $$z=\ln \tan\frac xy.$$ What is $z_x$ and what is $z_y$? Thanks ahead:) What I have tried: $$z_x=\frac{1}{\tan \frac xy} \frac{1}{1+(\frac xy)^2} \frac 1y=\frac {y}{\tan \frac xy (x^2+y^2)}$$ ...
0
votes
0answers
30 views

Understanding QEF's and assorted techniques for dual marching cubes

I'm attempting to implement dual marching cubes in a 3d engine and I'm getting lost in the math portion. http://www.cs.rice.edu/~jwarren/papers/dmc.pdf is the link to the original white paper. Part ...
1
vote
1answer
28 views

Can someone explain the concept of continuity and differentiability for functions of several variables?

Can someone explain the concept of continuity and differentiability for functions of several variables? Illustrated examples will definitely help, on how to solve problems(or establish proofs) of the ...
1
vote
0answers
28 views

Suppose limit as x approaches 0 of the derivative of f(x)=a. Show that the derivative at 0 exists and it is equal to a. [duplicate]

Let f be a continuous function on the interval [0,1], which is differentiable on (0,1). I know that I will have to use the definition of a derivative, which is the limit as x approaches 0 ...
4
votes
3answers
66 views

Find a continuous function on the reals where $f(x) >0$ and $f'(x) < 0$ and $f''(x) < 0$

We need to find a function $f(x)$ where $f(x) >0 $and $f'(x) < 0$ and $f''(x) < 0$ where $f$ is continuous for all real numbers. We have tried $ f(x) = \sqrt{-x}$ however this is not defined ...
4
votes
3answers
87 views

Functions $f$ such that $f(x+1)-f(x-1)=2f'(x)$.

What can one say about functions $f:\mathbb{R}\to\mathbb{R}$ satisfying the condition $f(x+1)-f(x-1)=2f'(x)$? Is is possible to find all such functions, or is this defining equation the best ...
1
vote
1answer
42 views

The chain rule, partial derivatives and general functions

I am revising for my first year Calculus examination. The following question is on a past paper and I am given the solution, however I am struggling to make sense of it: Let $V(x,y)$ be a ...
0
votes
1answer
54 views

How to find if and where $f(x)$ is continuous and/or differentiable for a given piecewise function? [closed]

What approach would be ideal in finding if and where $f(x)$ is continuous and/or differentiable for $ f(x) = \left\{ \begin{array}{lr} ln(x) & : x > e\\ \frac{x}{e} & : x \leq e ...
5
votes
1answer
94 views

Is the standard part function another devil's staircase?

The devil's staircase or Cantor function is an awesome function that increases value but has derivative zero everywhere (or "almost", whatever that means). I was incredibly amazed when I found out ...
-3
votes
3answers
120 views

Evaluate $\lim_{n\to\infty}({f(x+{1\over n})\over f(x)})$ [closed]

let $f$ be positive differentiable function defined on $(0,\infty)$. Then evaluate $\lim_{n\to\infty}({f(x+{1\over n})\over f(x)})$
1
vote
1answer
20 views

Graphing derivatives of non-function equations

Is it possible to graph the derivatives of equations that fail the vertical line test? Such as a circle, a folium of descartes, an asteroid, etc?
1
vote
1answer
42 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 ...
1
vote
1answer
39 views

Let P(x) be a polynomial of degree 4 , having extremum at $x=1,x=2$ and $\lim_{x\to 0}\frac{x^2+P(x)}{x^2}=2$ Then the value of P(2)

Let P(x) be a polynomial of degree 4 , having extremum at $x=1,x=2$ and $\lim_{x\to 0}\frac{x^2+P(x)}{x^2}=2$ Then the value of P(2) is _______ I worked out the limit using L'Hospital got a relation ...