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Question Number 220094 Answers: 1 Comments: 0

let n ≥ 2 ∈ Z and x_1 , x_2 , ..., x_n are a positive real numbers such that Σ_(i=1) ^n x_i = n , prove that Σ_(i=1) ^n (x_i ^n /(x_1 + ∙∙∙ + x_i ^ + ∙∙∙ + x_n )) ≥ (n/(n − 1))

$$ \\ $$$$\:\:\:\:\mathrm{let}\:{n}\:\geqslant\:\mathrm{2}\:\in\:\mathbb{Z}\:\mathrm{and}\:{x}_{\mathrm{1}} ,\:{x}_{\mathrm{2}} ,\:...,\:{x}_{{n}} \:\mathrm{are}\:\mathrm{a}\:\mathrm{positive}\:\mathrm{real}\:\mathrm{numbers}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{such}\:\mathrm{that}\:\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\:{x}_{{i}} \:=\:{n}\:,\:\mathrm{prove}\:\mathrm{that}\:\:\:\: \\ $$$$\:\:\:\:\:\:\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\:\frac{{x}_{{i}} ^{{n}} }{{x}_{\mathrm{1}} +\:\centerdot\centerdot\centerdot\:+\:\hat {{x}}_{{i}} \:+\:\centerdot\centerdot\centerdot\:+\:{x}_{{n}} }\:\geqslant\:\frac{{n}}{{n}\:−\:\mathrm{1}} \\ $$$$ \\ $$

Question Number 220072 Answers: 1 Comments: 0

If x,y∈(0,(π/2)) Then prove that: log_(sinx) ^2 (((sin2x)/(sinx + cosx))) + log_(cosx) ^2 (((sin2x)/(sinx + cosx))) ≥ 2

$$\mathrm{If}\:\:\:\mathrm{x},\mathrm{y}\in\left(\mathrm{0},\frac{\pi}{\mathrm{2}}\right) \\ $$$$\mathrm{Then}\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\mathrm{log}_{\boldsymbol{\mathrm{sinx}}} ^{\mathrm{2}} \:\left(\frac{\mathrm{sin2x}}{\mathrm{sinx}\:+\:\mathrm{cosx}}\right)\:+\:\mathrm{log}_{\boldsymbol{\mathrm{cosx}}} ^{\mathrm{2}} \:\left(\frac{\mathrm{sin2x}}{\mathrm{sinx}\:+\:\mathrm{cosx}}\right)\:\geqslant\:\mathrm{2} \\ $$

Question Number 220069 Answers: 1 Comments: 0

Let be (H_n )_(n≥1) H_n = Σ_(k=1) ^n (1/k) Find: lim_(n→∞) e^(2H_n ) ((((n+1)!))^(1/(n+1)) − ((n!))^(1/n) ) sin (π/n^2 ) = ?

Question Number 220022 Answers: 1 Comments: 0

If 0 ≤ a ≤ b ≤ 1 Then prove that: ∫_a ^( b) ∫_a ^( b) ∫_a ^( b) ((dxdydz)/( (√(1 + xyz)))) ≥ (b−a)^2 ∫_a ^( b) (dx/( (√(1 + x^3 ))))

Question Number 220020 Answers: 2 Comments: 0

If f:[a,b]→[−1,∞) a,b∈R a ≤ b f-continuous Then prove that: (∫_a ^( b) (1+f(x))dx)^3 ≥ (b−a)^3 + 3(b−a)^2 ∫_a ^( b) f(x)dx

$$\mathrm{If}\:\:\:\mathrm{f}:\left[\mathrm{a},\mathrm{b}\right]\rightarrow\left[−\mathrm{1},\infty\right) \\ $$$$\:\:\:\:\:\:\:\mathrm{a},\mathrm{b}\in\mathbb{R} \\ $$$$\:\:\:\:\:\:\:\mathrm{a}\:\leqslant\:\mathrm{b} \\ $$$$\:\:\:\:\:\:\:\mathrm{f}-\mathrm{continuous} \\ $$$$\mathrm{Then}\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\left(\int_{\boldsymbol{\mathrm{a}}} ^{\:\boldsymbol{\mathrm{b}}} \:\left(\mathrm{1}+\mathrm{f}\left(\mathrm{x}\right)\right)\mathrm{dx}\right)^{\mathrm{3}} \geqslant\:\left(\mathrm{b}−\mathrm{a}\right)^{\mathrm{3}} +\:\mathrm{3}\left(\mathrm{b}−\mathrm{a}\right)^{\mathrm{2}} \:\int_{\boldsymbol{\mathrm{a}}} ^{\:\boldsymbol{\mathrm{b}}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} \\ $$

Question Number 219957 Answers: 2 Comments: 0

if f:(0,∞)→(0,∞) f continuous and f((1/x)) + f((1/y)) = 2f((1/(x+y))) ∀ x,y > 0 then ∀ a,b > 0: ∫_a ^( b) ∫_a ^( b) ∫_a ^( b) f((1/(x+y+z)))dxdydz = (b−a)^2 ∫_a ^( b) f((1/x))dx

$$\mathrm{if}\:\:\:\mathrm{f}:\left(\mathrm{0},\infty\right)\rightarrow\left(\mathrm{0},\infty\right) \\ $$$$\:\:\:\:\:\:\mathrm{f}\:\:\mathrm{continuous} \\ $$$$\mathrm{and}\:\:\:\mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{x}}\right)\:+\:\mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{y}}\right)\:=\:\mathrm{2f}\left(\frac{\mathrm{1}}{\mathrm{x}+\mathrm{y}}\right) \\ $$$$\forall\:\mathrm{x},\mathrm{y}\:>\:\mathrm{0}\:\:\:\mathrm{then}\:\:\:\forall\:\mathrm{a},\mathrm{b}\:>\:\mathrm{0}: \\ $$$$\int_{\boldsymbol{\mathrm{a}}} ^{\:\boldsymbol{\mathrm{b}}} \int_{\boldsymbol{\mathrm{a}}} ^{\:\boldsymbol{\mathrm{b}}} \int_{\boldsymbol{\mathrm{a}}} ^{\:\boldsymbol{\mathrm{b}}} \:\mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{x}+\mathrm{y}+\mathrm{z}}\right)\mathrm{dxdydz}\:=\:\left(\mathrm{b}−\mathrm{a}\right)^{\mathrm{2}} \int_{\boldsymbol{\mathrm{a}}} ^{\:\boldsymbol{\mathrm{b}}} \mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{x}}\right)\mathrm{dx} \\ $$

Question Number 219956 Answers: 1 Comments: 0

let a,b,c,d > 1 f : [a , b] → [c , d] a continuous function for which ∃λ ∈ (a , b) such that a ∫_a ^( 𝛌) f(x) dx + b ∫_b ^( 𝛌) f(x) dx ≥ a + c then prove ∫_a ^( b) (x/(f(x))) dx ≤ ((1/a) + (1/b)) ((b^2 −a^2 −2)/2)

$$\mathrm{let}\:\:\:\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}\:>\:\mathrm{1} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\mathrm{f}\::\:\left[\mathrm{a}\:,\:\mathrm{b}\right]\:\rightarrow\:\left[\mathrm{c}\:,\:\mathrm{d}\right] \\ $$$$\mathrm{a}\:\mathrm{continuous}\:\mathrm{function} \\ $$$$\mathrm{for}\:\mathrm{which}\:\:\exists\lambda\:\in\:\left(\mathrm{a}\:,\:\mathrm{b}\right) \\ $$$$\mathrm{such}\:\mathrm{that} \\ $$$$\mathrm{a}\:\int_{\boldsymbol{\mathrm{a}}} ^{\:\boldsymbol{\lambda}} \:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{dx}\:+\:\mathrm{b}\:\int_{\boldsymbol{\mathrm{b}}} ^{\:\boldsymbol{\lambda}} \:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{dx}\:\geqslant\:\mathrm{a}\:+\:\mathrm{c} \\ $$$$\mathrm{then}\:\mathrm{prove} \\ $$$$\int_{\boldsymbol{\mathrm{a}}} ^{\:\boldsymbol{\mathrm{b}}} \:\frac{\mathrm{x}}{\mathrm{f}\left(\mathrm{x}\right)}\:\mathrm{dx}\:\leqslant\:\left(\frac{\mathrm{1}}{\mathrm{a}}\:+\:\frac{\mathrm{1}}{\mathrm{b}}\right)\:\frac{\mathrm{b}^{\mathrm{2}} −\mathrm{a}^{\mathrm{2}} −\mathrm{2}}{\mathrm{2}} \\ $$

Question Number 219940 Answers: 1 Comments: 0

Let: f : [n−1 , n] → [n , n + 1] be a continuous function Such that: ∫_(n−1) ^( n) (1 + xf^′ (x))dx ≤ nf(n)−(n−1)f(n−1) Then prove: ∫_(n−1) ^( n) (dx/(f(x))) ≤ (2/(n + 1)) , n∈N^∗

$$\mathrm{Let}: \\ $$$$\mathrm{f}\::\:\left[\mathrm{n}−\mathrm{1}\:,\:\mathrm{n}\right]\:\rightarrow\:\left[\mathrm{n}\:,\:\mathrm{n}\:+\:\mathrm{1}\right] \\ $$$$\mathrm{be}\:\mathrm{a}\:\mathrm{continuous}\:\mathrm{function} \\ $$$$\mathrm{Such}\:\mathrm{that}: \\ $$$$\int_{\boldsymbol{\mathrm{n}}−\mathrm{1}} ^{\:\boldsymbol{\mathrm{n}}} \left(\mathrm{1}\:+\:\mathrm{xf}\:^{'} \left(\mathrm{x}\right)\right)\mathrm{dx}\:\leqslant\:\mathrm{nf}\left(\mathrm{n}\right)−\left(\mathrm{n}−\mathrm{1}\right)\mathrm{f}\left(\mathrm{n}−\mathrm{1}\right) \\ $$$$\mathrm{Then}\:\mathrm{prove}: \\ $$$$\int_{\boldsymbol{\mathrm{n}}−\mathrm{1}} ^{\:\boldsymbol{\mathrm{n}}} \:\frac{\mathrm{dx}}{\mathrm{f}\left(\mathrm{x}\right)}\:\leqslant\:\frac{\mathrm{2}}{\mathrm{n}\:+\:\mathrm{1}}\:\:\:,\:\:\:\mathrm{n}\in\mathbb{N}^{\ast} \\ $$

Question Number 219935 Answers: 1 Comments: 0

Question Number 220100 Answers: 1 Comments: 0

Question Number 219899 Answers: 3 Comments: 0

Question Number 219898 Answers: 1 Comments: 0

Question Number 219884 Answers: 2 Comments: 0

(a,b,c)>0 such that, a+b+c=13, abc=36 find the maximum and minimum value of ab+bc+ca=?

$$\:\left({a},{b},{c}\right)>\mathrm{0}\:{such}\:{that}, \\ $$$$\:\:{a}+{b}+{c}=\mathrm{13},\:\:{abc}=\mathrm{36} \\ $$$$\:\:{find}\:{the}\:{maximum}\:{and}\:{minimum}\: \\ $$$$\:{value}\:{of}\:\:{ab}+{bc}+{ca}=? \\ $$

Question Number 219869 Answers: 1 Comments: 0

Question Number 219853 Answers: 1 Comments: 0

If 0<a≤b Then prove that ∫_a ^( b) (sinx)^(2sin^2 x) ∙ (1−sin^2 x)^(1−sin^2 x) dx ≥ ((b−a)/2)

$$\mathrm{If}\:\:\:\mathrm{0}<\mathrm{a}\leqslant\mathrm{b} \\ $$$$\mathrm{Then}\:\mathrm{prove}\:\mathrm{that} \\ $$$$\int_{\boldsymbol{\mathrm{a}}} ^{\:\boldsymbol{\mathrm{b}}} \left(\mathrm{sinx}\right)^{\mathrm{2}\boldsymbol{\mathrm{sin}}^{\mathrm{2}} \boldsymbol{\mathrm{x}}} \:\centerdot\:\left(\mathrm{1}−\mathrm{sin}^{\mathrm{2}} \mathrm{x}\right)^{\mathrm{1}−\boldsymbol{\mathrm{sin}}^{\mathrm{2}} \boldsymbol{\mathrm{x}}} \:\mathrm{dx}\:\geqslant\:\frac{\mathrm{b}−\mathrm{a}}{\mathrm{2}} \\ $$

Question Number 219846 Answers: 1 Comments: 0

If f:[a,b]→R 0<a≤b f - continuous Then prove that ((b^(4047) − a^(4047) )/(4047)) + ∫_a ^( b) f^2 (x^(2024) ) dx ≥ (1/(1012)) ∫_a^(2024) ^( b^(2024) ) f(x) dx

$$\mathrm{If}\:\:\:\mathrm{f}:\left[\mathrm{a},\mathrm{b}\right]\rightarrow\mathbb{R} \\ $$$$\:\:\:\:\:\:\mathrm{0}<\mathrm{a}\leqslant\mathrm{b} \\ $$$$\mathrm{f}\:-\:\mathrm{continuous} \\ $$$$\mathrm{Then}\:\mathrm{prove}\:\mathrm{that} \\ $$$$\frac{\mathrm{b}^{\mathrm{4047}} \:−\:\mathrm{a}^{\mathrm{4047}} }{\mathrm{4047}}\:+\:\int_{\boldsymbol{\mathrm{a}}} ^{\:\boldsymbol{\mathrm{b}}} \:\mathrm{f}\:^{\mathrm{2}} \:\left(\mathrm{x}^{\mathrm{2024}} \right)\:\mathrm{dx}\:\geqslant\:\frac{\mathrm{1}}{\mathrm{1012}}\:\int_{\boldsymbol{\mathrm{a}}^{\mathrm{2024}} } ^{\:\boldsymbol{\mathrm{b}}^{\mathrm{2024}} } \:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{dx} \\ $$

Question Number 219843 Answers: 0 Comments: 0

If a,b,c,d > 0 a^2 +b^2 +c^2 +d^2 = 4 Then prove that (1/((1+ab)^3 )) + (1/((1+ac)^3 )) + (1/((1+ad)^3 )) + (1/((1+bc)^3 )) + (1/((1+bd)^3 )) + (1/((1+cd)^3 )) ≥ (3/4)

$$\mathrm{If}\:\:\:\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}\:>\:\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} +\mathrm{c}^{\mathrm{2}} +\mathrm{d}^{\mathrm{2}} \:=\:\mathrm{4} \\ $$$$\mathrm{Then}\:\mathrm{prove}\:\mathrm{that} \\ $$$$\frac{\mathrm{1}}{\left(\mathrm{1}+\mathrm{ab}\right)^{\mathrm{3}} }\:+\:\frac{\mathrm{1}}{\left(\mathrm{1}+\mathrm{ac}\right)^{\mathrm{3}} }\:+\:\frac{\mathrm{1}}{\left(\mathrm{1}+\mathrm{ad}\right)^{\mathrm{3}} }\:+\:\frac{\mathrm{1}}{\left(\mathrm{1}+\mathrm{bc}\right)^{\mathrm{3}} }\:+\:\frac{\mathrm{1}}{\left(\mathrm{1}+\mathrm{bd}\right)^{\mathrm{3}} }\:+\:\frac{\mathrm{1}}{\left(\mathrm{1}+\mathrm{cd}\right)^{\mathrm{3}} }\:\geqslant\:\frac{\mathrm{3}}{\mathrm{4}} \\ $$

Question Number 219833 Answers: 0 Comments: 0