Questions tagged [absolute-value]
For questions about or involving the absolute value function also known as modulus function.
3,083
questions
3
votes
3
answers
72
views
Values of $m$ such that $|x^2-2x+m| + 2x + 1$ has $3$ extrema
I was given the following question:
Find the values of $m$ for which the curve $y=|x^2-2x+m| + 2x + 1$ has $3$ extrema.
My teacher suggested that we should use the quadratic formula $(b^2-4ac)$ and ...
- 31
0
votes
1
answer
118
views
Simplifying the expression $\frac{1}{3} \ln (x+2)^3+\frac{1}{2}\left[\ln x-\ln \left(x^2+3 x+2\right)^2\right]$
Express as a single logarithm. Simplify.
$$
\frac{1}{3} \ln (x+2)^3+\frac{1}{2}\left[\ln x-\ln \left(x^2+3 x+2\right)^2\right]
$$
So I am posting the question, how I solved it and then how the TA ...
- 33
-4
votes
1
answer
56
views
Evauate this limits without using $l´hopitals: \lim_\limits{x \to 0}\frac{x^2}{|x|}$
$$ \lim_\limits{x \to 0} \frac{x^2}{|x|}$$
my argument is that we are looking for values near zero, not in zero hence we can get rid of the x denominator with one in the numerator then evaluating the ...
- 95
0
votes
3
answers
61
views
Inequality involving nested absolute values
I don't understand where I'm wrong in my reasoning about this inequality:
$$\big\vert x - \vert 1-x\vert\big\vert < 1$$
Attempts
$$\vert 1-x\vert =
\begin{cases} 1-x & \text{if}\quad x\leq 1 \\ ...
- 2,418
-1
votes
0
answers
40
views
What is the abs max of x^2 on [-1, 1]?
If the abs Max of x^2 on (-1, 1) does not exist. Does that mean that the abs Max does or does not exist on [-1, 1]?
I understand that the abs min would be 0 for both (-1, 1) and [-1, 1], but I have a ...
- 15
1
vote
0
answers
28
views
proving concavity of the trigonometry function
I have a complicated function:
\begin{equation}
F(x,y) = \Big\lvert\cos(\sqrt{x^2 + 1}/2)\cos(\sqrt{y^2 + 1}/2) - \frac{xy + 1}{\sqrt{(x^2 + 1)(y^2+1)}} \sin(\sqrt{x^2 + 1}/2)\sin(\sqrt{y^2 + 1}/2) \...
- 77
1
vote
1
answer
46
views
doubt with absolute value
Assume a function
$$f(x,y) = 2\vert x \vert + 2\vert y\vert - \vert x-y \vert - \vert x+y \vert$$
If $x>y$ then $f(x,y) = 2y$,
If $x<y$ then $f(x,y) = 2\vert x \vert$
How to prove this?
If $x>...
- 113
8
votes
4
answers
2k
views
Why dividing equation by absolute value gives bad result
I have to solve $x^2 + 4x + 4 = 7|x+2|$.
I did this:
$(x + 2)^2 = 7|x+2|$
And we know that $|w| = w \iff w ≥ 0$, so:
$|x+2|^2 = 7|x+2|$ because the $(x+2)^2$ is always $≥0$
Then, I divided this ...
- 193
-1
votes
0
answers
82
views
Help with textbook problem. [closed]
Question: Sketch the graph of the following equation:
$ y = \sqrt{x+2} - \sqrt{x-2} $ (No Calculus)
Not sure where I would start with this one.
- 21
0
votes
0
answers
26
views
Prove that swapping values in sum of absolute differences produces a smaller result
I am stuck trying to mathematically prove the first statement that was made in the top answer to this question involving minimizing the sum of absolute differences between pairings of boys' and girls' ...
- 11
0
votes
2
answers
53
views
Quadratic Equation with $ |x|$ [duplicate]
Hello so I am trying to solve this problem:
$2x^2 - (a-4)|x| - a + 10 = 0 \ $
calculate every $a$, in which this equation has only $2$ solutions.
I tried to check two examples
$|x| \ge0$
$|x| < 0$
...
- 55
1
vote
2
answers
68
views
Trouble understanding the introduction of absolute-value inequalities in a proof
Prove that
$$f(x) = \frac{x^3-7x^2+6x+4}{x^2-5x+4}$$
is bounded on the the domain $D = \{x: 2 \le x \le 3\}.$
The answer key has the line
When $2 \le x \le 3$, we have $|x| < 3, 1 \le x-1 \le 2$ ...
- 21
0
votes
0
answers
64
views
What does the absolute value sign mean when used around the differential part of an integral?
The Wikipedia entry on Radon transform shows an equation like the following:
$$
\int_L f(\mathbf{x})|\mathbf{dx}|.
$$
What does absolute value around $$|\mathbf{dx}|$$ mean in an integral?
I ...
- 219
1
vote
0
answers
31
views
What does the Wirtinger derivative of a non-analytic function (absolute value squared) represent?
Let $z = x + i y$ be a complex number and consider the modulus squared function:
$$
f(z) = |z|^2 = z z^* = x^2 + y^2 = u(x, y)
$$
where the asterix denotes complex conjugation and $u(x, y)$ is the ...
- 11
3
votes
3
answers
156
views
Desmos and Wolfram Alpha contradict: Is $|1+x| \ge 1 - |x|$ true for all $x$?
Is $|1+x| \ge 1 - |x|$ true for all $x$?
When I solve this problem myself, I solved that it's true for all $x$.
But when I put it into Desmos, I get this:
https://www.desmos.com/calculator/vgxm3q1fso,...
- 31
-1
votes
1
answer
46
views
Integral of absolute difference of functions the same as integral of difference? [closed]
For some functions f and g, if
$$
\int f(x)-g(x) dx = \int g(x)-f(x)dx=c
$$
Then does
$$
\int |f(x)-g(x)| dx =c
$$?
I think this is true, however maybe it relies on some assumed symmetry?
- 1
1
vote
2
answers
197
views
If there is a negative operation in front of a grouping symbol, the terms inside must be multiplied by that operation, right?
I was cruising through my college algebra book, when I encountered this equation, everything is straight-forward, only, I have no idea what they did in regards to the grouping symbol, and the negative ...
- 29
1
vote
0
answers
93
views
Is my proof of the triangle inequality acceptable?
I want to show that $\forall a,b\in\mathbb{R}$, we have $|a|+|b|\ge|a+b|$, using the piecewise definition of the absolute value. In my attempt, I have broken the problem up into many cases. All of ...
- 420
1
vote
0
answers
48
views
Lower bound on expected value of folded normal [closed]
For $X \sim N(\mu, \sigma^2)$ the expected value of $|X|$ is calculated as
$$E|X| = \sigma \sqrt{\frac{2}{\pi}} e ^{-\mu^2/2\sigma^2} + \mu\left(1 - 2\Phi\left(-\frac{\mu}{\sigma}\right)\right)$$
I ...
- 49
1
vote
1
answer
76
views
Is my proof of the $|a||b|=|ab|$ property correct?
I want to show that $\forall a,b\in\mathbb{R}$, we have $|a||b|=|ab|$. Is the proof I have come up with correct? Have I accounted for every possible case?
Case 1, $a,b\ge0$:
By definition, $|a|=a$ and ...
- 420
0
votes
0
answers
69
views
Is there a "test" to see if a number is positive or negative?
This might be a silly question, but I'm curious if there is some sort of test to see if a number is positive or negative. What I mean by a test is that there is something that can be computed. For ...
- 1,921
0
votes
0
answers
39
views
$|\mathcal{o}(f(x))| \leq \mathcal{o}(|f(x)|)$
Let f(x) be a real continuous function
Is it true that
$|\mathcal{o}(f(x))| \leq \mathcal{o}(|f(x)|)$
thanks.
P.S.
$g(x)=\mathcal{o}(f(x))$ if $\lim \frac{g(x)}{f(x)}=0$
- 1,685
0
votes
1
answer
85
views
If $\sum a_n$ and $\sum b_n$ converge then is it true that$\sum\max\{a_n,b_n\}$ converges?
Let $$\sum a_n,\sum b_n$$ be convergent series and let $A_n,B_n$ be the $n\text{th}$ partial sums of $(a_n),(b_n)$ respectively. Let $c_n=\max\{a_n,b_n\}$ and let $C_n$ be the $n\text{th}$ partial sum ...
- 797
2
votes
1
answer
80
views
Strange simple inequality
In an online PDF about continuity the author claim (without proof) that for $h$ small and $f$ continuous we have:
$|f(x)-f(x+h)|<\dfrac{|f(x+h)|}{2}$
I try to prove if that statement is true or ...
- 211
0
votes
0
answers
145
views
Prove that: $|(x+y)(y+z)(z+x)| \le 1.$
Problem complex number inequality:
Given $x,y,z$ be complex numbers such that $|x|, |y|, |z| \le 1$ and $|x+y+z| \le 1.$ Prove that $$|(x+y)(y+z)(z+x)| \le 1.$$
I see it on AOPS here.
Here is my ...
- 315
0
votes
0
answers
35
views
How to find the minimum value of the given modulus equation using traingle inequality
My question is to find the minimum value of
$$|x-1|+|3x-1|+|5x-1|+|7x-1|+|9x-1|+|11x-1|$$
One way is to find the minimum value using graph by which I got the answer $X=\frac{1}{9}$
But I want to know ...
- 33
0
votes
4
answers
83
views
How to find minimum value of given multiple modulus function
So the question is to find the minimum value of given modulus function
$$f(x)=|x-1|+|3x-1|+|5x-1|+|7x-1|+|9x-1|+|11x-1|$$
So my first approach was to just make a graph of it
On making the graph the ...
- 33
0
votes
1
answer
47
views
Finding $y$ if more than two $x$ satisfy absolute value equation.
Consider the equation:
$$y=\big||x|-1\big|$$
In what interval will $y$ lie, if more than two values of $x$ should satisfy the above equation?
My work:
I was able to solve this by plotting the graph of ...
-1
votes
1
answer
49
views
Is the MVT true for $b<a$ (instead of $a<b$) ? Is this adapted absolute value version true?
The Mean Value Theorem for $f\in \mathbb R ^{\mathbb R}$, is stated only for $a < b$ as:
If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$ then there exists $x\in (a,b)$ such as: $$f(b)...
- 969
1
vote
0
answers
65
views
Algebraic rules of absolute value
In evaluating elementary $\epsilon$, $\delta$ proofs of limits, one often sees the following sort of move:
$$ \left|2x - 8\right| = \left|2(x-4)\right| = 2\left|x - 4\right| \dots$$ (See e.g. here (14:...
- 420
4
votes
1
answer
61
views
Why does the absolute value of difference between two consecutive integers in a finite string of positive reals always end with 0s?
Suppose I take a finite string of positive reals
1 4 19 3
In the first step, I find the absolute difference between consecutive numbers, the above string becomes
(4-1) (19-4) (19-3) (3-1) ⟹ 3 15 16 2
...
9
votes
1
answer
169
views
Can I replace modulus inequalities with rooted square arguements?
Suppose I want to show $|x-5|<|x+1|$. One way (and the way my lecturer shows) to do it is look at the negative and positive regions and solve the inequality. But with the definition $\sqrt{x}\geq0$,...
- 101
1
vote
1
answer
61
views
How to solve $xy'=3y-6x^2$ using integrating factors?
I am trying to solve $xy'=3y-6x^2$ using integrating factors. I am facing 2 issues when doing so. In order to find $P(x)$ to be used in $e^{\int P(x) dx}$, I am dividing by $x$ which, it seems, ...
- 1,701
3
votes
2
answers
85
views
for $x \in \mathbb{R}$ find the number of solutions in $3x^2 + 4|x^2 - 1| + x - 1 = 0$. Why we are considering negative values here?
I was solving some practice problems, from a booklet of math, and one of the problems is like this
for $x \in \mathbb{R}$ find the number of roots in the eq . $$3x^2 + 4|x^2 - 1| + x - 1 = 0$$
I know ...
- 235
2
votes
1
answer
44
views
Finding the derivative of a function with two absolute values within it, using a piecewise function
I'm trying to solve some a problem relating to absolute values. I found online the strategy for solving similar functions from here: https://www.youtube.com/watch?v=eIHtq67nh7w&list=...
1
vote
1
answer
69
views
The mode minimizes the $l_0$ norm?
Suppose we have a set $S$ of $N$ real numbers. Show that
$$\sum_{s_i\in S}|s_i-x|^0 $$
is minimal if $x$ is equal to the mode of S.
I'm a bit confused about that, because assuming $0^0 = 1$ the whole ...
- 2,622
0
votes
1
answer
45
views
Can I say $\frac{1}{n} \sum_{i}^{n} |x_i| \overset{P}{\to} \mathbb{E}(|X|)$ using LLN
I'm dealing with the following problem. Given $X_1,\ldots,X_n \sim^{iid}$ with density function:
$$
f_{X}(x) = \dfrac{1}{8} x e^{-\dfrac{|x|}{4}} \; \mathbb{I}_{(-\infty,\infty)} (x)
$$
Define: $ U_{...
1
vote
2
answers
122
views
Any polynomial with a positive leading coefficient is positive
I have some questions about this answer: https://math.stackexchange.com/a/2711058/1196218
Is $x$ positive (how can you tell?)? If not then im confused about a few things. For example, why would the ...
- 177
2
votes
1
answer
89
views
Can you pull out the sign-function out of the integral?
While playing aroud with integrals I stumbled across following identity:
$\newcommand{\sgn}{\text{sgn}}$
$\sgn(g(x))f(x) = \frac{d}{dx}\left(\int\sgn(g(x))f(x)dx\right) = \frac{d}{dx}\left(\sgn(g(x))\...
- 91
0
votes
0
answers
30
views
Archimedean absolute value on the set of real rational fractions, whose induced absolute value in $\mathbb{R}$ is the trivial one.
Archimedean absolute value on the set of real rational fractions, whose induced absolute value in $\mathbb{R}$ is the trivial one.
Can this be done?
So, my idea is to see it in $\mathbb{C}(t)$ so I ...
- 1,089
0
votes
0
answers
30
views
Is it true that $\sum_{i=1}^n \big|f(X_i) - \mathbb{E}[g(X_i)]\big| \leq \sum_{i=1}^n\big|f(X_i) - g(X_i)\big|$ on expectation?
Let $X_1,\ldots,X_n$ be i.i.d. random variables, and $f,g$ two functions. I am wondering whether the following result holds
$$
\mathbb{E}\left[\sum_{i=1}^n \big|f(X_i) - \mathbb{E}[g(X_i)]\big|\right]
...
- 63
0
votes
0
answers
15
views
What is the expected value for two complex numbers with known modulus and uniformly distributed phase?
It seems easy to prove that for $a=|a|e^{-j\arg(a)}$ and $b=|b|e^{-j\arg(b)}$, the expected value for the squared sum is $E\left[(a+b)^2\right]=|a|^2+|b|^2$. On the other side, the expected value $E\...
- 109
1
vote
1
answer
84
views
Why $\forall n \in \mathbb{Z}_{\geq 1}$ it is $n^{-1} \leq |n|_*$?
I was reading the proof of Ostrowski's theorem (which BTW is a beauty) and I got stumped here:
[O]ne has $|nr|_∗ \leq n|r|_∗$. [C]hoosing $r=n^{-1}$ shows that for all positive integer $n$, it holds $...
- 55
1
vote
1
answer
31
views
How could a hypothetical inverse to the absolute value function be represented?
So I know that $\int f'\left(x\right)\ dx$ would be $f\left(x\right)+C$ because for any value of C, $f'\left(x\right)$ would still be the same. Since it has infinite possibilities, we write a "+...
- 132
0
votes
0
answers
23
views
Find equation which gives you the profit for multiple trades
Automated market makers are protocols where you can exchange crypto tokens, they are based on pools which contain 2 tokens each pool, here is the definitions on how these pools work from Uniswap V2:
...
0
votes
3
answers
99
views
Solve a fractional equation
How do you go from this step:
$$
\frac{a_2b_2}{\left(b_2+x\right)^2}-\frac{a_1b_1}{\left(b_1-x\right)^2} = 0
$$
To this step:
$$
(a_1b_1-a_2b_2)x^2 + 2b_1b_2(a_1+a_2)x + b_1b_2(a_1b_2 - a_2b_1) = 0
$$...
0
votes
0
answers
14
views
Collapsed terminology for simple logarithmic procedure
During some computation, I had to rescale a value $x$ by taking the $\log_{10}(x)$. If $x$ was positive, I merely took the log, but if $x$ was negative i used -$\log_{10}(|x|)$. Is there a shorter ...
- 101
1
vote
1
answer
37
views
When is the absolute value of logarithm equal to the input?
So does the following equation have any closed form solution(s), or if not, can the solution(s) be expressed as functions of well known math constants (e.g., $\pi,e$)? Also, what is the relevance of ...
- 111
0
votes
0
answers
43
views
Trouble Calculating Integral of Absolute Value of Polynomials using Maxima and abs_integrate
I am currently working on a Maxima script that is supposed to calculate the definite integral from -1 to 1 of the absolute value ...
- 1
0
votes
1
answer
41
views
Why we squaring the norm of a complex function?
In Ginzburg Landau equation, there is a term,
$ |A|^2A$
and $A$ is space and time dependent function or $A(x,t)$
Why do we have norm or absolute value under square? Is square not enough?
My guess is ...
- 468