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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Is this inequality related to this integral correct?

Let $\alpha >0, 1/3 >\beta>0, T>0$, $h \in [0,T]$. In [Cerrai, Sandra. "Normal deviations from the averaged motion for some reaction–diffusion equations with fast oscillating ...
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Is that quantity bounded from below?

Let $p>1, s>0$ and $F$ be a function such that $$F(t)\le \frac{|t|^p}{p} +|t|^{p+p^{\prime}}e^{|t|^{p^{\prime}}}\quad\mbox{ for all } t\in\mathbb{R},$$ where $p^{\prime}$ denotes the conjugate ...
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How to proof Lip-constant weakens when size of function class approaches to infinity?

Taken from paper "A Universal Law of Robustness via isoperimetry" by Bubeck and Sellke. Theorem 3. Let $\mathcal{F}$ be a class of functions from $\mathbb{R}^{d} \rightarrow \mathbb{R}$ and ...
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1 vote
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Volume of the piece of an n-ball defined by inequality constraints

Consider $x \in \mathbb{R}^n$, and $B = \{x: ||x|| < r\}$, the n-ball with radius $r$ centered at the origin. Let $V$ be its volume. Further, consider $1 \leq m \leq n$ linear inequality ...
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How do I prove that $\forall a \forall b(\neg a<b\iff b\leq a)$ [duplicate]

The universe is the set of natural numbers including 0, which are defined in accordance with the Peano Axioms. We define the inequalities as: $a\leq b \iff \exists x(a+x=b)$ $a<b \iff \exists x(a+x=...
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Maximizing $ab+bc+cd+da$ when $a,b,c,d$ are positive real numbers, $abcd=7$, $a^2+b^2+c^2+d^2=20$ [closed]

Problem: maximizing $ab+bc+cd+da$ when $a,b,c,d$ are positive real numbers, $abcd=7$, $a^2+b^2+c^2+d^2=20$. I tried to solve that problem, but it is hard. The answer is $16$. How to solve that problem?...
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Is for a convex, continuous function f satisfying $Cf(s)f(t) \leq f(st) \leq Df(s)f(t)$ for some constants C>0 and D>0, implies that $f(t)=t^p$?

If $f(t)=t^p$, then f satisfies $Df(s)f(t) \leq f(st) \leq Df(s)f(t)$ for all s,t>0, for some constants C=1>0 and D=1>0. Can we get any other example other than $t^p$.
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Find number of continuous functions satisfying the equation $4\int_{0}^{\frac{3}{2}}f(x)dx+125\int_{0}^{\frac{3}{2}}\frac{dx}{\sqrt{f(x)+x^2}}=108$

The number of continuous functions $f:\left[0,\frac{3}{2}\right]\rightarrow (0,\infty)$ satisfying the equation$$4\int_{0}^{\frac{3}{2}}f(x)dx+125\int_{0}^{\frac{3}{2}}\frac{dx}{\sqrt{f(x)+x^2}}=108$$ ...
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Let $x,y,z\in[0,1]$. Find the maximum value of $\sqrt{|x-y|}+\sqrt{|y-z|}+\sqrt{|z-x|}$. [duplicate]

Let $x,y,z\in[0,1]$. Then find the maximum value of $\sqrt{|x-y|}+\sqrt{|y-z|}+\sqrt{|z-x|}$. Now the given answer is $2\sqrt{2}$ but I am not able to obtain the corresponding values of $x,y,z$. ...
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  • 7,082
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Let $X$ be an exponential random variable and $\Bbb{P}(X \in A) < \Bbb{P}(X \in B)$. Is $\Bbb{P}(aX \in A) < \Bbb{P}(aX \in B)$ for $a > 0$?

Let $X$ be an exponential random variable (say with mean 1) and $\Bbb{P}(X \in A) < \Bbb{P}(X \in B)$ for two events $A, B \subset\Bbb{R}$. Is $\Bbb{P}(aX \in A) < \Bbb{P}(aX \in B)$ for $a > ...
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4 votes
4 answers
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Is there a shorter or more trivial way to prove that $ x > \cos (x)-\cos (2 x) $ holds for all $x>0$?

I want to prove that the inequality $$ x > \cos (x)-\cos (2 x) $$ holds for all $x>0$. My attempt: Since the function on the RHS is periodic, we can find the position of extrema (on first ...
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Knowing $f(t^\ast)\ge 0$ (and some other information), can we show that $f(t)\ge 0$ at $t<t^\ast$?

I asked a similar question before and had to make several changes so before anyone spends time on answering it, I decided to clarify here. We have constants $\alpha_1,\alpha_2,r_1,r_2,c>0$ and we ...
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Evaluating the value of |$f'(x)$| [duplicate]

Let $f \in C^2[0,2]$ and $|f(x)| \le 1, |f''(x)|\le1$ for all $x \in[0,1]$ Prove that $|f'(x)|\le2 $ for all $x \in [0,2]$ I tried Taylor expansion to evaluate $|f'(x)|$, but it seems to work only ...
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2 votes
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Inequality of Narayana numbers

In some computations I am making, it is desirable to deduce that a certain expression is positive. I managed to rewrite it in terms of Narayana numbers, which are defined by $N(n,i) := \frac{1}{n}\...
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1 vote
1 answer
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Cauchy-Schwarz inequality and angle between two vectors

Notes I am reading these notes, and I can't understand the Cauchy-Schwarz inequality. It says that it proves that the input is between $[-1,1]$. The Cauchy-Schwarz inequality only states that the ...
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1 vote
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Proving an interesting basic inequality

I was reading an old paper in french where the authors claim that the following "obvious and natural inequality" holds. If $a_1,\ldots,a_n>0$ are all positive numbers, then $$\sum\limits_{...
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Taylor expansion in Hoeffding's Lemma proof

Hoeffding's Lemma proof use taylor expansion with this statement: From Taylor's theorem, for some $ 0\leq \theta \leq 1$ $ L(h) = L(0) + h L'(0) + \frac{1}{2} h^2 L''(h\theta) \leq \frac{1}{8}h^2 $ ...
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3 votes
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De Bruijn's inequality for series

According to Mitrinović, 3.9.42. page 359, N.G. De Bruijn proved the following result. Let $f$ be a decreasing and positive function such that $\sum_{n=1}^{+\infty} f(n)$ converges and let $a_{1}, \...
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Is my proof wrong or what could improved about $e^x\geq x+1$

Is the following proof true and new ? The proof : We suppose for $a,x>0$ fixed such that $a\geq 1 $ : $$e^{ax}\leq 1 +ax$$ using Bernoulli's inequality we have : $$1 +ax\leq (1+x)^a$$ We have : $$e^...
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0 answers
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Prove $(a+b)^{p} \leq a^{p}+b^{p}$ if $a,b>0$ and $p \in (0,1)$ [duplicate]

I need to prove that if $a,b>0$ and $p \in (0,1)$ then $(a+b)^{p} \leq a^{p}+b^{p}$. I've been trying to use the generalized binomial theorem but i haven't solve it yet.
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2 answers
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Find the value $\hat{\beta}$ which minimizes $\sum_{i=1}^{4}|i||i- \beta|$ for $\beta \in \mathbb{R}$

I am looking for the value $\hat{\beta}$ which satisfies the following condition: for each $\beta \in \mathbb{R}$, $$\sum_{i=1}^{4}|i||i - \hat{\beta}| \leq \sum_{i=1}^{4}|i||i - \beta|\text{.}$$ ...
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2 answers
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How to solve (not numerically) the inequality $x^a-x^b -c>0$?

Let $x>0$ and $a, b, c\in\mathbb{R}^*_+$ with $b>a$. I am wondering if there is a way (not numerical) to solve the inequality $$x^a-x^b -c>0.$$ Could someone please help or give some hints? ...
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Is this generalized metric $d_t(a,b):X\times X\longrightarrow [0,\infty)$ a continous function?

Consider the following generalized metric space where triangle inequality has been extended. Let $X$ be a non-empty set and $P, Q, R: X \times X \rightarrow[1, \infty)$. A function $d_{t}: X \times X \...
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Determine the sign of the expression with a concave function

Consider a concave function $f(x,y)$, which is defined for all $ x,y \in \mathbb{R_+}$. I want to determine under what condition(s) the sign of the following expression is negative: $$[f(x+\Delta x, y+...
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1 vote
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How to prove that there exists $C_1>0$ such that $ \max\{\|f_{n+1}-f_n\|_{\infty}, \|g_{n+1}-g_n\|_{\infty}\}\le 2^nC^nC_1^n\frac{T^n}{n!} $?

I have two iterated update upper bound for two sequences of continuous functions $\{f_n\}$ and $\{g_n\}$, that is there exists $C>0$ such that $$ \|f_{n+1}-f_n\|_{\infty}\le C\int_0^T|f_n(s)-f_{n-1}...
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  • 1,808
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Showing inequality for the norm of an integral equation

Let $\xi \in \mathbb{R}^2$, $\Phi \in C^0([0, +\infty[, \mathbb{R}^{2x2})$ a bounded function. Let $y:[0, +\infty[ \rightarrow \mathbb{R}^2$ a solution of the following integral equation, $$ y(x) = e^{...
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5 votes
1 answer
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Elementary proof that $\text{Re}\big(\frac{z^{n+1} - n z - z + n}{(z-1)^2}\big) \ge \frac{n}2$?

I would like an elementary proof that the real part of $$ f(z) = \frac{z^{n+1} - n z - z + n}{(z-1)^2} $$ is greater than or equal to $n/2$ for any $z \in \mathbb C$, $|z| \le 1$, where $n$ is a ...
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-1 votes
0 answers
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A bound on a function involving exponentials [closed]

Does there exist constants $c,x_0>0$ such that $$\sqrt{e^{1/x}-1}\left(1-\sqrt{e^{-1/x}}\right)\leqslant \frac{c}{x^2}$$ for all $x>x_0$? So far I have only been able to show that this is $O(x^{...
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3 votes
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Prove that $(k+2)^\alpha \sum\limits_{i=1}^{k}\frac{1}{(1+i)^\alpha} \ge (k+1)^\alpha \sum\limits_{i=1}^{k+1}\frac{1}{(1+i)^\alpha}$

In this problem, I'm trying to prove that for any $\alpha\in[1/2,1]$, there exists a constant $K(\alpha)$ such that for any $k\geqslant K(\alpha)$, the inequality $$(k+2)^\alpha\cdot\left(\frac{\sum_{...
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4 votes
0 answers
88 views

Inequality with special case of equality at $\infty$

Prove that if $a,b,c,d$ are positive reals we have: $$\frac{a}{\sqrt{a^2+b^2}}+\frac{b}{\sqrt{b^2+c^2}}+\frac{c}{\sqrt{c^2+d^2}}+\frac{d}{\sqrt{d^2+a^2}}\leq3$$ I think that I have found a equality ...
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2 votes
1 answer
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Solve: $\sec(2x) \ge\sec(x) , x\in [0,\pi]$\ {$\frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}$}

Here is the following question: Solve: $\sec(2x) \ge\sec(x) , x\in [0,\pi]$\ {$\frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}$} Note: This is part b of a question where in part a, I was asked to solve: $...
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4 votes
0 answers
51 views

Inequality with interesting independent constants

Let $b_1,\dots,b_{n-1}$ be integers satisfying $0 \le b_i \le n-i$ for each $i \in [n-1]$ such that $\sum_{i=1}^{n-1} b_i = \alpha \binom{n}{2}$ where $\alpha$ is constant strictly between $0$ and $1$....
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0 answers
28 views

How can i find minimum value of this inequality

$a_1,a_2,...,a_n$ are 8 distinct positive integers. $b_1,b_2,...,b_n$ are another 8 distinct positive integers ($a_i,b_j$ are not necessarily y distinct for $i, j = 1, 2, ...8$).Enter the smallest ...
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2 votes
1 answer
53 views

How do we rigorously prove that for $n>1$, $(1+x)^{n-1}<1$ for $-1<x<0$?

Given $n>1$ and $$(1+x)^{n-1}<1$$ Intuitively I can see that for $x \in (-1,0)$, we have $1+x<1$, and if we raise that to any power then it will be smaller than 1. How do we prove this ...
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  • 1,715
2 votes
0 answers
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If $\sum_{cyc}\frac{a}{a+1}=1$ Show that $abc\le \frac{1}{8}$

Given positive reals $a,b$ and $c$, such that $$\sum_{cyc}\frac{a}{a+1}=1$$ Show that $$abc\le \frac{1}{8}$$ This problem is fairly easy we can just clear denominators and then use AM-GM. But since I ...
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-1 votes
0 answers
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A cyclic inequality with fractions [closed]

I want to show $\frac{1}{a+b-ab}+\frac{1}{a-b+ab}+\frac{1}{-a+b+ab}\geq \frac{3}{2}$ for $a,b \geq 1$ directly. Could you give me an advice? I think we have to note that the denominators are not ...
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-2 votes
1 answer
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If $-x>|y|>z$, then how does $x+y$ compare to $|y|+z$?

If $-x>|y|>z$, then how does $x+y$ compare to $|y|+z$? an absolute value is non-negative, so $x<0$ and $z\geq 0.$
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1 vote
1 answer
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If two functions are close apart can I prove the difference of their empirical loss is also small?

I am trying to understand the proof of Theorem 3 in the paper "A Universal Law of Robustness via isoperimetry" by Bubeck and Sellke. Basically there exist atleast one $w_{L,e}$ in $\...
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1 vote
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How to the estimate as application of Strichartz Estimates?

RecallA pair $(q,r)$ is admissible if $q\geq 2, r\geq 2$ and $\frac{2}{q} = N \left( \frac{1}{2} - \frac{1}{r} \right),(q,r,N)\neq(\infty,2,2).$ Strichartz estimates Let $\phi \in L^2(\mathbb R^N),$...
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1 vote
1 answer
21 views

If $−1<x<4$ then determine $a$ and $b$ in $a<2x+3<b$.

In this problem: "If $−1 < x < 4$ then determine $a$ and $b$ in $a < 2x + 3 < b$" I solved it like this: $-1 < x < 4$, Interval notation for $x$: $(-1, 4)$ $a < 2x + 3 &...
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-1 votes
1 answer
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Corrected conjecture about a possible inequality $\sum_{i=1}^{n}\sqrt{\frac{x_i+1}{4x_i^2+10x_i+4}}\leq \frac{n}{3}$ .

Hi it's a follow up of Prove $\sum_{cyc}{\sqrt{\frac{x+1}{x^2+16x+1}}}\geqslant 1$ and $ \sum_{cyc}{\sqrt{\frac{x+1}{4x^2+10x+4}}}\leqslant 1$ for $x,y,z>0,xyz=1$ : Problem : Let $x_i>0$ and $n$ ...
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0 votes
0 answers
13 views

How can I show a simple inequality? [duplicate]

I was looking for some paper, and I had a question about simple inequality process. How can we show that following inequality? $\sum_{t=1}^{T}\frac{1}{t}\leq 1+\log{T}$ For those who want a detailed ...
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2 votes
1 answer
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Inequality involving integral of 1/log(t)

I want to show that $ \forall x > 1$; $ x \ln(2) \leq \int_x^{x^2} \frac{1}{\ln(t)}dt \leq x^2 \ln(2) $ I've tried tp: study monotony of the fuction $\int_x^{x^2} \frac{1}{\ln(t)}dt - x^2 \ln(2)$. ...
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1 vote
0 answers
20 views

How to get the Lipschitz constant $L$ using the inequality?

Taken from paper "A Universal Law of Robustness via isoperimetry" by Bubeck and Sellke. $$ \begin{aligned} &\mathbb{P}\left(\exists f \in \mathcal{F}: \frac{1}{n} \sum_{i=1}^{n}\left(y_{...
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4 votes
2 answers
97 views

An interesting recurrent equality, possibly easier to solve in its differential form?

I encountered an interesting inequality that I'm not sure how to approach. Here $c$ is a positive constant. $$f(n+1) - f(n) = c f(n)\sum_{m=0}^n f(m)$$ I am not familiar with techniques to solve ...
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  • 475
0 votes
1 answer
19 views

Simple word problem about inequalities

Person $A$ can perform a task in $x$ seconds, while person $B$ takes $x+2$ seconds to do the same. When $A$ and $B$ collaborate, they manage to complete it in less than $t$ seconds. What is the ...
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2 votes
2 answers
66 views

An inequality: from the complex to the real case.

Let $p\in(1,2]$ and $q\in[2,\infty)$ be its conjugate exponent, then for $z,w\in\mathbb{C}$ the following inequality holds $$ \Large \left|\frac{z+w}{2}\right|^q+\left|\frac{z-w}{2}\right|^q\leq\left[\...
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0 votes
0 answers
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Question about $H^1$ norm

The $H^1$ norm is given by: $$(u,v)_{H^1}=\int_0^1u′v′+\int_0^1uv$$ But in a problem that I'm solving, I have a constant in the second integral: $$|a(u,v)|\le|k|||u'||_{L^2}||v||_{L^2}+\Bigg|\int_0^1u'...
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  • 11
3 votes
1 answer
46 views

$0 \leqslant a \leqslant b \Rightarrow \|a\| \leqslant \|b\|$ in a $C^*$-algebra

Let $A$ be a $C^*$-algebra and $a,b\in A$. Therefore $0\leqslant a \leqslant b \Rightarrow \|a\|\leqslant \|b\|$. I'm trying to prove this claim, but apparently it's necessary to use some spectral ...
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2 votes
3 answers
87 views

Prove $\sqrt{x-1} + \sqrt{y-1} \le xy, x \ge 1, y \ge 1$

Let $x$ and $y$ be real numbers, such that $x \ge 1$ and $y \ge 1$. Prove that this inequality is true: $\sqrt{x-1} + \sqrt{y-1} \le xy$ Can someone show me steps to solve it. PS:I need to give steps ...
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