0
votes
1answer
39 views

How to show $f(x,y) \leq \theta f(x,y) + (1-\theta)f(x,y)$ for $\theta \in [0,1]$?

Let $\theta \in [0, 1]$. Let $f(x,y)$ be a function. Is there a way I could prove that $f(x,y) \leq \theta f(x,y) + (1-\theta)f(x,y)$? I have tried to start with $f(x,y) = 2f(x,y) - f(x,y)$ or ...
2
votes
2answers
30 views

Establish the absolute maximum of a function

We have this function:$$f(x)=\begin{cases} \sin(x) \cdot\ln(\sin2x), & \mbox{if }0<x<\pi/2 \\ 0, & \mbox{if }x=0,\mbox{or }x=\pi/2 \end{cases}$$ So, how to prove that it decreases and ...
2
votes
4answers
113 views

Proving exponential is growing faster than polynomial

Let $P(x)$, a polynomial which isn't the zero-polynomial. I want to prove the following limits $$\lim \limits_{x\to\infty} \left|P(x)\right|e^{-x} = 0$$ $$\lim \limits_{x\to-\infty} ...
2
votes
2answers
49 views

Examples of interesting integrable functions with at least 2 fixed points and an explicit inverse

What are some interesting functions I can use to demonstrate this integration trick: $$\int_a^b [f(x)+f^{-1}(x)]=b^2-a^2$$ I would like to know of some interesting functions where this trick is not ...
1
vote
3answers
48 views

Is Inverse Function Composition Commutative?

Given $f: \mathbb{R}\to(-1,1)$ is there a theorem that states $f\left(f^{-1}(x)\right) = f^{-1}\left(f(x)\right)$. In example, is $\tanh{\left(\tanh^{-1}{(x^2)}\right)} ...
1
vote
2answers
33 views

How to check the function is convex or not?

How one can check the function whether it is convex or not. I know one method by using Hessian Matrix but I think it did not fit for the following example. I think Hessian matrix method cannot be ...
3
votes
0answers
36 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) $x^{2}f^{\prime}(x)=f^{2}(f(x)),~\forall x\in\lbrack0,\infty)$
3
votes
2answers
42 views

Composition of functions is constant in $\mathbb{R^2}$.

Let $\hspace{0.05cm}f:\mathbb{R^2}\to\mathbb{R^2}$ $\hspace{0.05cm}$ and $\hspace{0.05cm}$$g:\mathbb{R^2}\to\mathbb{R^2}$ $\hspace{0.05cm}$ be such that$\hspace{0.05cm}$ $g\circ f$$\hspace{0.05cm}$ ...
0
votes
2answers
20 views

Prove for a monotone function, if $x_0$ is interior on interval $I$ then the one-sided limits exists

The proof goes like this: Lets suppose $f$ is nondecreasing (for nonincreasing we'll observe the function $-f(x)$). Let $x_0$ an interior point in the interval $I$, and $\left\{x_k\right\}$ an ...
0
votes
4answers
66 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
1answer
40 views

Understanding a theorem regarding to monotonic functions

Let $f$ be a monotonic on an interval $I$. if $x_0$ is interior to $I$, then the one-sided limits $\lim_{x\to {x_0}^-}f(x)$ and $\lim_{x\to {x_0}^+}f(x)$, both exists. Suppose the theorem is ...
-2
votes
2answers
38 views

Graph of floor function [closed]

Please help me to draw the graph of $f(x)= \lfloor{x^2-1}\rfloor$. Please give me some tips.
1
vote
0answers
29 views

How to establish the equivalence of these two statements about integrals of step functions?

First Statement: Let $s$ be an arbitrary step function defined on the closed interval $[a, b]$. Then we have $$ \int_{ka}^{kb} s\left(\frac{x}{k}\right) \ dx = k \int_a^b s(x) \ dx $$ for every $k ...
0
votes
1answer
31 views

How to establish this equivalence for integrals of step functions?

First Statement: Let $s$ be an arbitrary step function defined on the closed interval $[a,b]$. Then we have $$\int_{a}^{b} s(x) \ dx = \int_{a+c}^{b+c} s(x-c) \ dx.$$ Second Statement: Let $s$ be ...
1
vote
1answer
47 views

Using 4 step-rule $y = 2/ (4t - 3)^{2}$ [closed]

I tried solving it. My answer is $-4/16t^{2} + 48t + 18$, if your answer is different kindly show how is it done too thanks
2
votes
1answer
39 views

Greatest value of f

If $f'(x)=6-x$ then which of the following has the greatest value? $f(2.01)-f(2)$ $f(3.01)-f(3)$ $f(4.01)-f(4)$ $f(5.01)-f(5)$ $f(6.01)-f(6)$ I know the answer is $f(2.01)-f(2)$ but how to prove?
1
vote
3answers
56 views

True or False Question About Functions [closed]

If $f(1)>0$ and $f(3)<0$, then there exists a number $c$ between $1$ and $3$ such that $f(c)=0$. I'm not sure how to solve this question. Thanks in advanced!
1
vote
1answer
33 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 ...
0
votes
0answers
37 views

L'Hospital's rule for higher derivatives

Let $u,v \in C^\infty(\mathbb{R})$, where $u(0) = 0$ and $v(0) = 0$ and $v'(0) \not= 0$. Then, one can define a function $f \in C^\infty(\mathbb{R}\setminus\{0\})$ by $f := u/v$. L'Hospital allows ...
3
votes
3answers
42 views

Finding domain of $f\text{ o }g$

I am having a small question, please don't close this before answering, I just want to know whether its a matter of convention or not. If $f(x) = \dfrac{1}{x}$ and $g(x) = \dfrac{1}{x}$ $ $ Then ...
-1
votes
1answer
53 views

Find $k$ so that $f(x)$ is a continuous function [closed]

Find $k$ so that $f(x)$ is a continuous function. $$f(x)=\left\{\begin{array}{ll}x^2 &x\leq2\\ k-x^2 & x>2 \end{array}\right.$$ Does anyone know how to go about this problem? Thanks in ...
2
votes
1answer
49 views

Let $f$ be continuous and $U \subset \mathbb{R}^n$ open, if $f: U \rightarrow \mathbb{R}^m $ is injective then $n \leq m$?

I had intended to restrict the image then $f:U \rightarrow f(U) \subset \mathbb{R}^m $ is bijective. Therefore $\dim f(U) = n \leq m$. That's right?
0
votes
5answers
172 views

How to prepare this function for integration

I want to prepare $$f(x)=\frac{x}{1+x^2}$$ for integration, how do i get the $1+x^2$ to the top? Is $$\frac{x}{1+x^2}$$ the same as $\frac x1 + \frac{x}{x^2}$? If not please explain how I prepare the ...
1
vote
1answer
55 views

What distribution is this?

Top: Uniform, Bottom: ?? Distribution. Ignore the random spikes - those are just binning errors. Looking for a distribution that is on $[0,1]$ and is equal to $0$ at $1$ and some positive ...
1
vote
0answers
63 views

Find function such that $\displaystyle f(1)=10 \ , \ f'(0)=-2$ and $f(x) >0 \ \ \forall x \in \mathbb{R}$

I'm trying to find a function under the following conditions: $f(1)=10$ $f'(0)=-2$ $f'(x)$ is monotonically decreasing. I want to find a function such that $\displaystyle f(x)>0 \ \forall x \in ...
1
vote
2answers
64 views

Can all functions be expressed in terms of elementary functions?

After being introduced to the non-elementary function through an attempt to evaluate $\int x \tan (x)$, an interesting question occurred to me: Can the non-elementary functions be decomposed to ...
1
vote
1answer
25 views

Derive property from continuity - is this proof valid?

Prove that if $f:R^+ \rightarrow R^+$ is continuous on the positive reals and is decreasing, then for all $a$ there exists an $\eta > 0$ such that $(a-\eta)f(a-\eta) > \frac{1}{2}a*f(a)$. EDIT ...
1
vote
1answer
52 views

Find $\lim_{x\to-1} f(x)$ for $f(x) = (x^2 - 2x - 3) / (x+ 1)$

I need to find the following limit: $$\lim_{x\to -1}\frac{x^2 - 2x - 3 }{x + 1}$$ The polynomial is simplified to $\dfrac{(x+1)(x-3)}{x+1}$ Hello, I can solve this by plugging in the value $-1$ ...
0
votes
2answers
45 views

Period of $\frac{\sin(Ny)}{sin y}$ with $N$ odd?

The function $$f(y) = \displaystyle \frac{\sin(Ny)}{\sin y}$$ is periodic with period $2 \pi$ in general. But tracing the graphic of that function for $N$ odd it seems that for $0 \leq x < \pi$ ...
0
votes
4answers
74 views

Can a limit of a function be not an integer?

I'm just taking calc, and all the teacher's examples gave only integer results. Is it possible to have fractions or decimals?
1
vote
3answers
41 views

How to establish these two facts about polynomials?

Let $f(x) := \sum_{k=0}^n c_k x^k $ be a polynomial of degree $n\geq 0$ with real coefficeints such that $f(x) = 0$ for $n+1$ distinct real values of $x$. Then how to prove that each $c_k = 0$ and ...
3
votes
2answers
39 views

$|f'(x)|\le|f(x)|$ and $f(0)=0$, prove that $\forall x\in[0,\frac 1 2]:f(x)=0$

Let $f:[0,\frac 1 2]\to \mathbb R$ be differentiable and let $|f'(x)|\le|f(x)|$ and $f(0)=0$. Prove that $\forall x\in[0,\frac 1 2]:f(x)=0$ I got stuck when I tried to solve this. If we'll ...
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} ...
1
vote
2answers
54 views

Spivak's Calculus - Chapter 5 Problem 8

(For making things simple - everywhere where I've written "$\lim f(x)$" I meant $\lim_{x\to a} f(x)$) I'm trying to do this excercise: And apparently the answers in the answer book are yes, yes, ...
-1
votes
0answers
17 views

Linear Equation Formulas for specific questions

Im trying to figure out how to do this problem, but it is just extremely confusing to understand how too do. A cricket chirps at different retes depending on temperature. You can estimate the ...
4
votes
2answers
66 views

$f'$ is bounded and isn't continuous on $(a,b)$, so there's a point $y\in(a,b)$ such that $\lim_{x\to y}f'$ does not exist

Prove/disprove: $f$ has a bounded derivative and $f'$ isn't continuous on $(a,b)$, so there's a point $y\in(a,b)$ such that $\displaystyle\lim_{x\to y}f'$ does not exist. I think that if $f'$ ...
0
votes
0answers
12 views

Confusion about quasi-concavity

If I have a function such as $y=(x^a + y^b)^2$ with $a$ and $b$ both greater than one... is it enough to show that it is not quasiconcave by showing that the second derivatives are not negative? The ...
2
votes
1answer
28 views

Question about the local maxima of a funciton

Assume $f(x_1,x_2,\dots,x_n)$ is a smooth, continuos, differentiable function, and let we want to check if $(x'_1,x'_2,\dots,x'_n)$ is the local maxima or not. Assume the first order condition is ...
1
vote
4answers
91 views

Roots of $x \cos{x}-1$ and $\cos{x}-x^{-1}$

To finding the roots of $x \cos{x}-1=0$ we can write the equation as $$ f(x)=x \cos{x}-1=0 \to x \cos{x}=1 \to \cos{x}=\frac{1}{x} \to \cos{x}-\frac{1}{x}=u(x)-v(x)=g(x)=0 $$ The roots of $f(x)$ are ...
1
vote
3answers
56 views

Show $f$ is uniformly continuous

Let $f$ continuous function on $[0,\infty)$. Lets assume there are $a,b$ such that: $\lim_{x\rightarrow \infty} f(x)-(ax+b) = 0$. Prove $f$ is uniformly continuous on $[0,\infty)$. Well, At ...
1
vote
1answer
106 views

What is integral of $x^x$?

I have no idea on how to approach this problem. I tried solving it by taking logarithm and then evaluating, but that won't serve the purpose I guess. Can someone please help?
0
votes
4answers
43 views

What is $f$? Finding where a function converges pointwise?

I have a question. Let $f_n(x) = x^{4n} + \frac1{n^2}$. AS $n \to \infty$, $f_n$ converges pointwise to a function $f$ on $[0,1]$ What is $f$? Now if I am understanding correctly, couldn't ...
2
votes
2answers
69 views

$\exists x_0$ such that $f(f(x_0))=x_0$ prove that $f$ has a fixed point

Let $f:\mathbb R\to \mathbb R$ be coninuous. Suppose there exists $x_0$ such that $f(f(x_0))=x_0$. Prove that $f$ has a fixed point or in other words: $\exists c\in\mathbb R: f(c)=c$ . Suppose ...
0
votes
1answer
46 views

$f$ is differentiable twice, bounded and has a minimum on $x_0$, prove that there's a point $y\in\mathbb R$ such that $f''(y)=0$

Suppose $f:\mathbb R\to \mathbb R$ is differentiable and there's a constant $c>0$ such that $f'(x)>c$ for all $x\in(a,\infty)$. Prove that $\displaystyle\lim_{x\to\infty}f(x)=+\infty$ ...
1
vote
1answer
81 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
42 views

Handy way to find the $x$ value where $\sin x \cos \left( \frac{\pi}{2} \sin x \right)$ is maximum?

Like in the title, is there a handy way to compute the $x$ values for which the function $$f(x) = \sin x \cos \left( \frac{\pi}{2} \sin x \right)$$ reaches its maxima? The derivative is $$f'(x) = ...
2
votes
0answers
36 views

Fractional derivative of exponential function

With the $n$th order derivative ($n$ as a positive integer) of $e^{ax}$ given by $$D^{n}e^{ax}=a^ne^{ax},$$ is the generalized (or fractional) derivative the same? Does it apply for any arbitrary ...
5
votes
1answer
108 views

There cannot exist a rational function $f: \mathbb{R} \to \mathbb{R}$ injective, not surjective

I was looking for a rational function $f: \mathbb{R} \to \mathbb{R}$ that looks like $\arctan$, in that it is injective not surjective well-defined on all $x\in \mathbb{R}$ (no vertical ...
7
votes
4answers
166 views

A function satisfying $f(\frac1{x+1})\cdot x=f(x)-1$ and $f(1)=1$?

$f:[0,\infty)\to\mathbb{R}$ is a continuous function which satisfies $f(1)=1$ and: $$f(\frac1{x+1})\cdot x=f(x)-1$$ Does there exist such a function, if they do, are there infinitely many? And is ...
2
votes
1answer
36 views

Taylor series $\ln(2+x)$ centered at $x=2$

Taylor series $\ln(2+x)$ centered at $x=2$. Is the correct result $$y=\ln \left(4\right)+\sum _{n=1}^{∞}\frac{\left(-1\right)^n}{4^{\left(2^{\Large n}\right)}}\cdot \frac{\left(x-2\right)^n}{n!}\ ?$$ ...