Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities. Do not use this tag just because an inequality appears somewhere in your question.

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Ukrainian tough inequality for university

Prove that for every $x\ge0$ and natural numbers N, $(x+1)^{n+1}>ne x^n.$ I rearranged to get $\left(\frac{\left(x+1\right)}{x}\right)^{1+n}>\frac{ne}{x}$ not sure how to continue.
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How to prove an inequality by induction [closed]

I am having difficulty proving this by induction: $$\sum_{k=1}^n \frac{k}{k^2 +1} \le \frac{n}{2}$$ Any tips on how to approach the problem? Thank you!
elguero's user avatar
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Does a constant $C>0$ exist such that for $\forall\ p\in\mathbb{C}[z_1,z_2]$ we have: $\sup_{z\in rD^2}|p(z_1,z_2)|\le C\sup_{z\in D^2}|p(z_1,z_2)|$?

Question: Does a finite constant $C>0$ exist such that for $\forall\ p\in\mathbb{C}[z_1,z_2]$ we have: $$\sup_{z\in r \mathbb D^2}|p(z_1,z_2)|\le C\sup_{z\in \mathbb D^2}|p(z_1,z_2)|$$ where $r>...
anon's user avatar
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A tail bound similar to Markov inequality

I am trying to prove the following: If $Z$ is s.t. $\mathbb{E}(Z)=0$ and $\mathbb{E}(Z^2)=1$, for any $r >0$, $$\mathbb{P}(Z\geq r)\leq (1 + r^2)^{-1}.$$ I tried a lot using Markov and Chebyshev ...
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An inequality with a "Cauchy-Schwarz" flavour

Let $x_1\leq x_2\leq ... \leq x_n$ and $y_1<y_2< ... <y_n$ be positive integers such that $x_1\geq 2$, $x_i < y_i$ and $y_i + 1<y_{i+1}$. Do we have that $$x_ny_n (\sum_{i=1}^n x_i)^2 \...
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How to prove $\lim_{n\to \infty} (1+\frac{1}{1!}+\frac{1}{2!}+...\frac{1}{n!}) = e$ with $x_n = (1+\frac{1}{n})^n$?

I'm very confused about the question below, which I couldn't figure out for days. In Example 5 the author is teaching us proving $$\lim_{n\to \infty} (1+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+...\frac{...
John HHU's user avatar
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How to prove this inequality related to complex function?

$f: \mathbb{D} \rightarrow \mathbb{C}$ is analytic with $f(0)=0, f^{\prime}(0)=1$, and satisfies that $|f(z)|<R$ on $\mathbb{D}$ for some $R>1$. $$ r := \inf \{|w|: w \notin f(\mathbb{D})\} . $$ ...
KY LIAO's user avatar
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Given a system of n inequalities, prove the following [closed]

x ^ 4 - 2alpha_{l} * x ^ 3 + (alpha_{l} ^ 2 - 2alpha_{l} + 2) * x ^ 2 - 2alpha_{l}*x + (alpha_{l} - 1) ^ 2 <= 0 where each alpha t in [1/2, 5] (i=1,2,3...n). Let x_{l} be an arbitrary solution ...
Jay Khandelwal's user avatar
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31 views

Prove that $(a + b + c)(ab + bc + ca)(a^3 + b^3 + c^3) \le (a^2 + b^2 + c^2)^3$. [duplicate]

Problem: Let $a$, $b$, and $c$ be positive real numbers. Prove that $$(a + b + c)(ab + bc + ca)(a^3 + b^3 + c^3) \le (a^2 + b^2 + c^2)^3.$$ I have tried expanding everything out, and applying Muirhead’...
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Inequality question(wish to confirm working)

Let $a_{i}$ be a set of positive real numbers such that $\sum_{i=1}^n a_{i}^3=3$ and $\sum_{i=1}^n a_{i}^5=5$. Prove that $\sum_{i=1}^n a_{i} > \frac{3}{2}$. My attempt: Using Titu's lemma, $$\frac{...
A shubh's user avatar
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3 votes
2 answers
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If $x>0,y>0$ and $4xy=2^{x+y}$ then find the minimum and maximum values of $x+y$.

If $x>0,y>0$ and $4xy=2^{x+y}$ then find the minimum and maximum values of $x+y$. My Attempt I tried by putting $t=x+y$ $\Rightarrow 4x(t-x)=2^t$. On differentiation we have $4t-8x=\frac{dt}{dx}(...
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Relationship between variance and covariance

I know that $$\text{var}(x-y) = \text{var}(x) + \text{var}(y) - 2\text{cov}(x,y)$$ and $$\text{cov}(x,y) = \frac{1}{2}(\text{var}(x) + \text{var}(y) - \text{var}(x-y)).$$ Is it possible to say that $$\...
lela's user avatar
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0 answers
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Prove that $\mathop{max}_{x\in [0,1]}|u(x)|\le \frac{1}{8}\mathop{max}_{x\in [0,1]}|{u''(x)}|$ when $u(x)\in C^2[0,1],u(0)=u(1)=0$

I want to prove the inequality with analysis methods instead of method of PDE Actually we can consider the following PDE: $$\left\{\begin{array}{l} u''(x)=u''(x)\\ u(0)=u(1)=0 \end{array}\right.$$ ...
Gang men's user avatar
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Question about proof on inequalities

I have four equations: $n\geq 5k+2$ $n_1\geq 5k_1+1$ $n_2\geq 5k_2 +1$ $n=n_1+n_2$ What I want to get eventually is $k_1+k_2=k$, but it seems like I cannot say that as I get $n\geq 5(k_1+k_2)+2$ and ...
Johny's user avatar
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3 votes
3 answers
134 views

Elementary proof that $(1+1/x)^x$ is increasing for $x \in \mathbb{R}_{>0}$

All similar proofs I could find show that $(1+1/n)^n$ is increasing for positive integers values of $n$ only, or show that the derivative of $(1+1/x)^x$ is positive for all $x \in \mathbb{R}_{>0}$. ...
Hussein Aiman's user avatar
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1 answer
80 views

Convex function derivatives inequality

Decide whether there exists function $f: \mathbb{R} \to (0,+\infty)$ such that for all $x \in \mathbb{R}$ we have $f''(x)f(x)>(f'(x))^{2}$. I know that if such function would exist then $f''(x) &...
Math_man's user avatar
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2 votes
7 answers
78 views

Showing that for every acute angle $x$ in a right triangle $\frac{1}{\sin x}+\frac{1}{\cos x}\ge 2\sqrt 2$ is always true

The problem is to show that in a right-angle triangle with hypothenuse K and sides M and N, the inequality $\frac{K}{M}+\frac{K}{N}\ge 2\sqrt 2$ is always true. My approach: I tried to simplify the ...
Billy's user avatar
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Exercise 8.15 Brezis - Interpolation inequality

I have a problem with this exercise (see the text in the following link). Interpolation like inequality ,Question from Brezis' book exercise 8.15 The link practically solves it. Only one last step ...
Seurat's user avatar
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1 answer
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why this equality involving matrix holds true?

I am studying Lemma 11 of the paper I am having difficulty understanding on the last step, in particular, I have two questions: The first question (1) On the first equality of the last step on page 15,...
chloe's user avatar
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-2 votes
2 answers
44 views

Log summation with geometric progression

Problem statement: Show that $\ln\left(x_{n}\right)<\sum_{k=1}^{n}\frac{1}{3^{k}}$ where $x_{n}$ = $\left(1+\frac{1}{3}\right)\left(1+\frac{1}{3^{2}}\right)...\left(1+\frac{1}{3^{n}}\right)$ I ...
LÜHECCHEgon's user avatar
2 votes
1 answer
27 views

Inequality of entropies for Bernoulli plus Gaussian

Question: Given $X\sim\text{Bernoulli}(\alpha)$, $Y\sim\mathcal{N}(0,1)$, and two non-random constants $C_1$ and $C_2$ such that $C_1>C_2$. What can we say about the inequality between the two ...
Resu's user avatar
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Subgaussian concentration implies bound on probability density

Very simple question: Suppose $X$ is a subgaussian RV with a density $p_X(x)$. Does this imply any nontrivial upper bounds on $p_X(x)$ for large $x$? Some context: It is not hard to show that bounds ...
student566's user avatar
-1 votes
0 answers
55 views

How to prove why the boundary of this inequality is 0.5? [closed]

I have to prove why the boundary of this inequality is 0.5.It is known as birthday attack boundary I wonder how to do this. Can anyone help? $$ \prod_{i\ =0}^{2^{n-1}}\left(1-\frac i{2^n}\right)\le\...
niloufar hz's user avatar
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1 answer
26 views

Inequality regarding inner product and functions with zero integral

Suppose $X$ is a finite set and $f:X\rightarrow \mathbb{R}$ satisfies $\sum_{x\in X}f(x)=0$. Let $p\in\Delta(X)$ be a probability measure on $X$. Does the following statement hold? $$ \sum_{x\in X} f(...
Lemma1's user avatar
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2 votes
2 answers
124 views

Prove $(\Re{z_1})^2+(\Re{z_2})^2+(\Re{z_3})^2 < \frac{3}{2}$ for solutions of $4z^3-4z^2+12z-1=0$

The statement of the problem : Let $z_1, z_2, z_3$ be the solutions of the equation $$4z^3-4z^2+12z-1=0$$ Prove that : $(\Re{z_1})^2+(\Re{z_2})^2+(\Re{z_3})^2 < \frac{3}{2}$ ($\Re x$ is the real ...
Last X's user avatar
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1 vote
0 answers
41 views

Suppose functions $y$ and $f$ are non-negative and continuous on the interval, prove this inequality

Suppose functions $y$ and $f$ are non-negative and continuous on the interval $<a,b>$ (where $<$ can be $( $ or $[ $ and $>$ can be $] $ or $) $), and let $\lambda > 0$. If the ...
Arbatus's user avatar
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0 votes
1 answer
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How to prove $a^{b}+b^{a} \geq 1$ for all $a, b \geq 0$? [duplicate]

It is clear if $a \geq 1$ or $b \geq 1$, but how can I show it when $0< a,b <1$? I want hints.
Mahmoud albahar's user avatar
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0 answers
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Upper bound on matrix vector multiplication

For a matrix $A$, let $\|A\|_2$ denote the spectral norm (operator norm with euclidean norm used in both spaces). And for a vector $x$, let $\|x\|_2$ denote the euclidean norm. Is the following ...
Dylan Dijk's user avatar
-2 votes
0 answers
38 views

Upper Bound of a product of sines

Consider the function $$ f_n(t)= \prod_{1 \leq k \leq n-1,\\gcd(k,n)=1} \sin(t-\frac{k \pi}{n}),t \in [0,\pi].$$ I wonder whether it is possible to compute some nontrivial uppper bounds of the ...
AgnostMystic's user avatar
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0 answers
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Estimating Difference of Two Averages of Function by Its BMO Norm

Assume $f\in \text{BMO}(\mathbb{R}^d)$. Denote $f_Q=\displaystyle\dfrac{1}{|Q|}\int_Q f(x)\,dx$. I want to show for $\alpha>2$ and any cube $Q$ with positive volume, we have $|f_{\alpha Q}-f_Q|\leq ...
Laurence PW's user avatar
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0 answers
29 views

Bounds for Gamma function expression

Let $a,b,c,d \in \mathbb R$ such that $c\ge 1$ and $0 < d < 2$. Let $i$ be the imaginary unit and $\Gamma$ the Euler Gamma. Numerically it looks that $$2^{(a+bi)}\Gamma\left(\frac{c-(a+bi)}{d} \...
zelda's user avatar
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1 vote
0 answers
40 views

Prove the following inequality for $a,b,c,d>0$.

Let $a,b,c,d>0$. Is there an easy way to see that the following inequality is true: $$4(a+b+c+d)+2(a^2+b^2+c^2+d^2)-4(ab+ac+ad+bc+bd+cd)-(a^2b+ab^2+a^2c+ac^2+a^2d+ad^2+b^2c+bc^2+b^2d+bd^2+c^2d+cd^2)...
Ryan Hendricks's user avatar
4 votes
3 answers
157 views

If $x,y,z>0$ and $x+y+z=1$,then find the maximum value of $(1-x)(2-y)(3-z)$.

If $x,y,z>0$ and $x+y+z=1$,then find the maximum value of $(1-x)(2-y)(3-z)$. My Attempt: We have $0<1-x<1,1<2-y<2,2<3-z<3$ so A.M-G.M inequality cannot be used since we can never ...
Maverick's user avatar
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1 vote
0 answers
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Minimize $P=\sqrt{a^2b+b^2c+c^2a+abc}+\sqrt{a+b+c-1}$ when $ab+bc+ca=3.$

Let $a,b,c\ge 0:ab+bc+ca=3.$ Find the minimal value $$P=\sqrt{a^2b+b^2c+c^2a+abc}+\sqrt{a+b+c-1}.$$ I think $2+\sqrt{2}$ is desired result. $P$ attain this value when $a=b=c=1.$ I use $a+b+c\ge \sqrt{...
Hello world's user avatar
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Inequality with Products and Sums

I need help to find a proof for the following inquality. Assuming that $ 0 \leq c_i \leq 1 $ and $ 0 \leq d_i \leq 1 $, show that $$ \prod_{i=1}^N (c_i + d_i - c_i d_i) \geq \prod_{i=1}^N c_i + \prod_{...
Duns's user avatar
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1 answer
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p-variation of x

Let us consider data $(k \in \mathbb{Z}_{\ge 1}, (s_1, \ldots, s_k) \in \mathbb{R}^k_{\ge 0})$ such that $s_1 + \ldots + s_k = 1$. The mesh of a datum like that is $\max (s_1, \ldots, s_k)$. Given $p &...
Sasha's user avatar
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2 votes
2 answers
73 views

The total number of non-negative integers $n$ satisfying the equations $n^2=p+q$ and $n^3=p^2+q^2$, where $p$ and $q$ are integers, is...

The total number of non-negative integers $n$ satisfying the equations $n^2=p+q$ and $n^3=p^2+q^2$, where $p$ and $q$ are integers, is... Case$1$: If both $p$ and $q$ are non-negative integers then, ...
aarbee's user avatar
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3 votes
0 answers
61 views

A self-proof of Vapnik - Chervonenkis theorem

Theorem: For every $\varepsilon >0$, with the probability greater than $1-\varepsilon$ \begin{align*} R_p(\hat{g}_{n,\mathcal{G}}) - R_{p}(g^*_{p,\mathcal{G}}) \le 2 \sqrt{\dfrac{2V_{\mathcal{G}...
Eto's user avatar
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1 answer
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I'm wondering if we can prove the following inequality with those ideas.

We aim to prove that $\tan\left(\frac{|x+y|}{1+|x+y|}\right)$ $\leq \tan\left(\frac{|x|}{1+|x|}\right) + \tan\left(\frac{|y|}{1+|y|}\right)$ for all $x, y \in \mathbb{R}$. Let's assume that $\tan\left(...
impact21's user avatar
2 votes
1 answer
73 views

Prove that $\int f \ln(f) d \mu =\sup \left \{ \int f \phi d \mu : \int e^{\phi} d\mu \leq 1 \right \}$

Question Prove that $\int f \ln(f) d \mu = \sup \left \{ \int f \phi d \mu : \int e^{\phi} d\mu \leq 1 \right \}$ With $f$ verifying $ \int f d \mu = 1 $ and $ f \cdot \ln(f) $ is integrable, with $ \...
OffHakhol's user avatar
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3 votes
1 answer
255 views

Proof of Triangle Inequality for $d(g; x, y) = \left(|x-y|^4 + g\,| x \times y |^2\right)^{\frac{1}{4}}$

I am seeking assistance in proving that a function, denoted as $d(g; x, y)$, defined on $\mathbb{R}^2 \times \mathbb{R}^2$ and parameterized by the non-negative real number $g$, may satisfy the ...
roiban12096's user avatar
-4 votes
0 answers
55 views

A very old problem on Maxima and minima [closed]

Find the maximum value of the expression $$ |\left|x_1-x_2\right|-x_3\left|-\ldots-x_{1990}\right|, $$ where $x_1, x_2, \ldots, x_{1990}$ are distinct natural numbers between 1 and 1990. (O Bogopol'...
SquïdÆir's user avatar
1 vote
1 answer
71 views

Prove that $a+b+c+d \geq \frac{2}{a+1}+\frac{2}{b+1}+\frac{2}{c+1}+\frac{2} {d+1}$

the question Let $a,b,c,d>0$ with $a+b+c+d \geq \frac{1}{a}+ \frac{1}{b}+\frac{1}{c}+ \frac{1}{d}$. Prove that $a+b+c+d \geq \frac{2}{a+1}+\frac{2}{b+1}+\frac{2}{c+1}+\frac{2} {d+1}$ the idea Maybe ...
IONELA BUCIU's user avatar
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7 votes
2 answers
231 views

Prove ${2n \choose n+i} \geq e^{-8 i^2/n} {2n \choose n}$

I am trying to prove ${2n \choose n+i} \geq e^{-8 i^2/n} {2n \choose n}$ for $0\leq i \leq n$. My attempt: I rewrote ${2n \choose n+i}$ to $${2n \choose n+i} = {2n \choose n} \prod_{1\leq j \leq i} \...
AspiringMat's user avatar
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0 answers
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Searching for a more concise solution to $|x - 1| + |x + 1| < 2$ [duplicate]

I came up with what I think is the solution to exercise 11. (v) on chapter 1 of the third edition of book Calculus by Michael Spivak. Find all numbers $x$ for which $|x - 1| + |x + 1| < 2$. ...
Approxiz's user avatar
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0 answers
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A modified version of log-sum inequality.

Suppose $a_i>0, b_i>0, c_i>0 \; \forall i = 1, 2, \dots, n$, and $$\sum_{i} a_i b_i \ln(\frac{c_i}{b_i}) \geq 0, $$ where $a_i, b_i, c_i$ are not constant over $i$. Moreover, $$\sum_{i}a_ib_i\...
entropy's user avatar
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0 votes
1 answer
90 views

Prove that $f(u+v)\le f(u) +f(v)$ being $f(u)= Au\cdot u\in\mathbb R$

Let $A\in\mathbb R^{n\times n}$ be a symmetric and positive definite matrix. Consider the function $$f: u\in\mathbb R^n\mapsto f(u)= (Au\cdot u)^{1/2}\in\mathbb R.$$ Let $\|\cdot\|_1$ denote a norm ...
C. Bishop's user avatar
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2 votes
2 answers
108 views

How to prove $\sum\limits_{cyc} \frac{1}{x+yz} \le \frac{9}{2(xy+yz+zx)}$ for all $x,y,z >0:x+y+z=3.$?

When I entered a test at my school, I stuck this problem (it is also posted here) Let $x,y,z$ be positive real numbers such that $x+y+z=3$, prove that $$\frac{1}{x+yz}+\frac{1}{y+zx}+\frac{1}{z+xy} \...
30 Anh Ti 711's user avatar
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0 answers
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How to prove $\frac{a}{a’}<\frac{a+b+c+d}{a’+b’+c’+d’}<\frac{d}{d’}$ [closed]

Given $a, b, c, d, a', b', c', d'>0$ and $\displaystyle\frac{a}{a’}<\frac{b}{b’}<\frac{c}{c’}<\frac{d}{d’}$ , prove that $\displaystyle\frac{a}{a’}<\frac{a+b+c+d}{a’+b’+c’+d’}<\frac{...
Moli's user avatar
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0 answers
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A functional inequality over integers

Given positive integers $N$ and $n$, we define a function $f(x)$ as follows: $$f(x):=\frac{x^2-3x}{2}~\text{ when } x\leq N,$$ and $$f(x):=\frac{1}{2n}x^2+\frac{n-4}{2}x~\text{ when } x>N.$$ ...
Kim's user avatar
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