Mathematics

Questions tagged [tangent-line-method]

For proofs inequalities by Tangent Line method.

Let $f:\mathbb{R}\rightarrow\mathbb R$ be differentiable function, $\sum\limits_{k=1}^nx_k=n$ and we need to prove that $\sum\limits_{k=1}^nf(x_k)\geq nf(1)$.

Thus, $$\sum\limits_{k=1}^nf(x_k)-nf(1)=\sum\limits_{k=1}^n\left(f(x_k)-f(1)\right)=\sum\limits_{k=1}^n\left(f(x_k)-f(1)+\lambda(x_k-1)\right),$$ where $f'(1)+\lambda=0$.

137 questions
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Prove $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$

Let $a,b,c$ are non-negative numbers, such that $a+b+c = 3$. Prove that $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$ Here's my idea: $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$ $2(\sqrt{a} + \sqrt{b} + \sqrt{c}) \ge 2(ab + bc + ca)$…
Stefan4024
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13
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3 answers

Proving $\left(\frac{a}{a+b+c}\right)^2+\left(\frac{b}{b+c+d}\right)^2+\left(\frac{c}{c+d+a}\right)^2+\left(\frac{d}{d+a+b}\right)^2\ge\frac{4}{9}$

The inequality: $$\left(\frac{a}{a+b+c}\right)^2+\left(\frac{b}{b+c+d}\right)^2+\left(\frac{c}{c+d+a}\right)^2+\left(\frac{d}{d+a+b}\right)^2\ge\frac{4}{9}$$ Conditions: $a,b,c,d \in \mathbb{R^+}$ I tried using the normal Cauchy-Scharwz, AM-RMS, and…
Mathejunior
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Prove that if $a+b+c+d=4$, then $(a^2+3)(b^2+3)(c^2+3)(d^2+3)\geq256$

Given $a,b,c,d$ such that $a + b + c + d = 4$ show that $$(a^2 + 3)(b^2 + 3)(c^2 + 3)(d^2 + 3) \geq 256$$ What I have tried so far is using CBS: $(a^2 + 3)(b^2 + 3) \geq (a\sqrt{3} + b\sqrt{3})^2 = 3(a + b)^2$ $(c^2 + 3)(d^2 + 3) \geq 3(c + d)^2$…
Sandel
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4 answers

Given that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$.Show that:$(a+1)(b+1)(c+1)\ge 64$

Given that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$ show that: $$(a+1)(b+1)(c+1)\ge 64$$ My attempt: First I tried expanding the LHS getting that $$abc+ab+bc+ca+a+b+c \ge 63$$ I applied Cauchy-Schwarz on $(a+b+c)(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})$…
Spasoje Durovic
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A inequality proposed at Zhautykov Olympiad 2008

An inequality proposed at Zhautykov Olympiad 2008. Let be $a,b,c >0$ with $abc=1$ . Prove that: $$\sum_{\mathrm{cyc}}{\frac{1}{(a+b)b}} \geq \frac{3}{2}.$$ Set $a=\frac{x}{y}$ , $b=\frac{y}{z}$ , $c=\frac{z}{x}$ . Our inequality becomes:…
Iuli
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Prove or disprove $\sum\limits_{1\le i < j \le n} \frac{x_ix_j}{1-x_i-x_j} \le \frac18$ for $\sum\limits_{i=1}^n x_i = \frac12$($x_i\ge 0, \forall i$)

Problem 1 : Let $x_i \ge 0, \, i=1, 2, \cdots, n$ with $\sum_{i=1}^n x_i = \frac12$ . Prove or disprove that $$\sum_{1\le i < j \le n} \frac{x_ix_j}{1-x_i-x_j} \le \frac18.$$ This is related to the following problem: Problem 2 : Let $x_i \ge 0, \,…
River Li
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2 answers

Inequalities - Tangent line trick

I will first state the "trick": we fix $a=\frac{a_1+a_2+...+a_n}{n}, \ $ If $f$ is not convex we can sometimes prove: $$f(x)\ge f(a)+f'(a)(x-a) $$ If this manages to hold for all x, then summing up the inequality will give us the desired conclusion.…
Spasoje Durovic
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Prove that $\sum \limits_{cyc}\frac {a}{(b+c)^2} \geq \frac {9}{4(a+b+c)}$

Given $a$, $b$ and $c$ are positive real numbers. Prove that:$$\sum \limits_{cyc}\frac {a}{(b+c)^2} \geq \frac {9}{4(a+b+c)}$$ Additional info: We can't use induction. We should mostly use Cauchy inequality. Other inequalities can be used rarely.…
user2838619
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Prove that the minimum values of $x^2+y^2+z^2$ is $27$ with given condition $\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=1$.

Question: Prove that the minimum values of $x^2+y^2+z^2$ is $27$ , where $x,y,z$ are positive real variables satisfying the condition $\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=1$ . From AM $\ge$ GM, we have $\left( \dfrac{x^2+y^2+z^2}{3}\right)^3\ge…
Primo Raj
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1 answer

Proof of an interesting inequality

I think this question was asked here before, but I am unable to find it at the moment. Apologies if this is due to my ineptitude. Anyway, the question is as follows: let $n>1$ be an integer number and $a_1,\dots,a_n\in\mathbb{R}^+$ . We define…
Leo163
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Monotonicity of the function $(1+x)^{\frac{1}{x}}\left(1+\frac{1}{x}\right)^x$.

Let $f(x)=(1+x)^{\frac{1}{x}}\left(1+\frac{1}{x}\right)^x, 0<x\leq 1.$ Prove that $f$ is strictly increasing and $e<f(x)\leq 4.$ In order to study the Monotonicity of $f$, let $$g(x)=\log f(x)=\frac{1}{x}\log (1+x)+x\log…
Riemann
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Using Rearrangement Inequality .

Let $a,b,c\in\mathbf R^+$ , such that $a+b+c=3$ . Prove that $$\frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}\ge\frac{a^2+b^2+c^2}{2}$$ $Hint$ : Use Rearrangement Inequality My Work :- $\\$ Without Loss of Generality let's assume that $0\le a\le…
arnav_de
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A more elementary proof that if $x_i>0$ for $1\leq i\leq n$, and $\sum x_i=1$, then $(x_1+\frac{1}{x_1})\cdots(x_n+\frac{1}{x_n})\geq(n+\frac1n)^n$

For $x_i>0$ , $1\leq i\leq n$ and $\sum_i x_i=1$ , show that $$\left(x_1+\frac{1}{x_1}\right)\cdots \left(x_n+\frac{1}{x_n}\right)\geq \left(n+\frac{1}{n}\right)^n$$ I think this can be proved easily by Jensen's inequality. However, my child…
maomao
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3 answers

Prove that $ a^2+b^2+c^2 \le a^3 +b^3 +c^3 $

If $ a,b,c $ are three positive real numbers and $ abc=1 $ then prove that $a^2+b^2+c^2 \le a^3 +b^3 +c^3 $ I got $a^2+b^2+c^2\ge 3$ which can be proved $ a^2 +b^2+c^2\ge a+b+c $ . From here how can I proceed to the results? Please help me to…
Chris
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5 answers

Minimize $\frac{2}{1-a}+\frac{75}{10-b}$

Let $a,b>0$ and satisfy $a^2+\dfrac{b^2}{45}=1$ . Find the minimum value of $\dfrac{2}{1-a}+\dfrac{75}{10-b}.$ WA gives the result that $\min\left(\dfrac{2}{1-a}+\dfrac{75}{10-b}\right)=21$ with $a=\dfrac{2}{3},b=5$ . Consider making a…
WuKong
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