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Question Number 150056 Answers: 2 Comments: 1

Question Number 150053 Answers: 0 Comments: 0

let x;y;z;t>0 and x+y+z+t=4 prove that (4/((xyzt)^2 )) + 3 ≥ (√(45 + x^4 + y^4 + z^4 + t^4 ))

$$\mathrm{let}\:\:\mathrm{x};\mathrm{y};\mathrm{z};\mathrm{t}>\mathrm{0}\:\:\mathrm{and}\:\:\mathrm{x}+\mathrm{y}+\mathrm{z}+\mathrm{t}=\mathrm{4} \\ $$$$\mathrm{prove}\:\mathrm{that} \\ $$$$\frac{\mathrm{4}}{\left(\mathrm{xyzt}\right)^{\mathrm{2}} }\:+\:\mathrm{3}\:\geqslant\:\sqrt{\mathrm{45}\:+\:\mathrm{x}^{\mathrm{4}} \:+\:\mathrm{y}^{\mathrm{4}} \:+\:\mathrm{z}^{\mathrm{4}} \:+\:\mathrm{t}^{\mathrm{4}} } \\ $$

Question Number 150045 Answers: 0 Comments: 0

Question Number 150044 Answers: 2 Comments: 0

Given f(((2x−3)/(2x+1)))+f(((2x+3)/(1−2x)))= 4x f(x)=?

$$\:\mathrm{Given}\:\mathrm{f}\left(\frac{\mathrm{2x}−\mathrm{3}}{\mathrm{2x}+\mathrm{1}}\right)+\mathrm{f}\left(\frac{\mathrm{2x}+\mathrm{3}}{\mathrm{1}−\mathrm{2x}}\right)=\:\mathrm{4x} \\ $$$$\:\mathrm{f}\left(\mathrm{x}\right)=? \\ $$

Question Number 150040 Answers: 0 Comments: 0

Let a,b,c be positive real numbers such that a+b+c=1 .Prove that ((ab)/(1−c^2 )) +((bc)/(1−a^2 ))+((ca)/(1−b^2 )) ≤ (3/8)

$$\mathrm{Let}\:\mathrm{a},\mathrm{b},\mathrm{c}\:\mathrm{be}\:\mathrm{positive}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{such} \\ $$$$\mathrm{that}\:\mathrm{a}+\mathrm{b}+\mathrm{c}=\mathrm{1}\:.\mathrm{Prove}\:\mathrm{that}\: \\ $$$$\:\frac{\mathrm{ab}}{\mathrm{1}−\mathrm{c}^{\mathrm{2}} }\:+\frac{\mathrm{bc}}{\mathrm{1}−\mathrm{a}^{\mathrm{2}} }+\frac{\mathrm{ca}}{\mathrm{1}−\mathrm{b}^{\mathrm{2}} }\:\leqslant\:\frac{\mathrm{3}}{\mathrm{8}} \\ $$

Question Number 150039 Answers: 1 Comments: 0

Prove that (a/b)+(b/c)+(c/a)≥(√((a^2 +1)/(b^2 +1)))+(√((b^2 +1)/(c^2 +1)))+(√((c^2 +1)/(a^2 +1))) for a,b,c are positive real number

$$\mathrm{Prove}\:\mathrm{that}\:\frac{\mathrm{a}}{\mathrm{b}}+\frac{\mathrm{b}}{\mathrm{c}}+\frac{\mathrm{c}}{\mathrm{a}}\geqslant\sqrt{\frac{\mathrm{a}^{\mathrm{2}} +\mathrm{1}}{\mathrm{b}^{\mathrm{2}} +\mathrm{1}}}+\sqrt{\frac{\mathrm{b}^{\mathrm{2}} +\mathrm{1}}{\mathrm{c}^{\mathrm{2}} +\mathrm{1}}}+\sqrt{\frac{\mathrm{c}^{\mathrm{2}} +\mathrm{1}}{\mathrm{a}^{\mathrm{2}} +\mathrm{1}}} \\ $$$$\mathrm{for}\:\mathrm{a},\mathrm{b},\mathrm{c}\:\mathrm{are}\:\mathrm{positive}\:\mathrm{real}\:\mathrm{number}\: \\ $$

Question Number 150037 Answers: 0 Comments: 0

Random Problem: ∫_(π/4) ^(π/2) (−7sin x + 3cos x) dx By getting the antiderivative of the trigonometric functions: ∫ sin(x) dx = −cos x + c ∫ cos(x) dx = sin x + c = −7 ∫ sin x + 3 ∫ cos x ∣_(π/4) ^(π/2) = −7(− cos x) + 3(sin x) ∣_(π/4) ^(π/2) = 7 cos x + 3sin x ∣_(π/4) ^(π/2) Evaluate it to the top and bottom limit of integration: = (7 cos ∙ (π/2) + 3 sin ∙ (π/2))− (7 cos ∙ (π/(4 )) + 3 sin ∙ (π/4) ) =[7(0) + 3(1)] − [7(((√2)/2)) + 3(((√2)/2))] = 3 − ((7(√2))/2) − ((3(√2))/2) = 3 − ((10(√2))/2) or 3 − 5(√2) Answer: 3 − 5(√2) Solution by Roswel:)

$${Random}\:{Problem}: \\ $$$$\underset{\frac{\pi}{\mathrm{4}}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\:\left(−\mathrm{7sin}\:{x}\:+\:\mathrm{3cos}\:{x}\right)\:{dx} \\ $$$$ \\ $$$${By}\:{getting}\:{the}\:{antiderivative}\:{of}\:{the}\:{trigonometric}\:{functions}: \\ $$$$\int\:\mathrm{sin}\left({x}\right)\:{dx}\:=\:−\mathrm{cos}\:{x}\:+\:{c} \\ $$$$\int\:\mathrm{cos}\left({x}\right)\:{dx}\:=\:\mathrm{sin}\:{x}\:+\:{c} \\ $$$$=\:−\mathrm{7}\:\int\:\mathrm{sin}\:{x}\:\:+\:\:\mathrm{3}\:\int\:\mathrm{cos}\:{x}\:\underset{\frac{\pi}{\mathrm{4}}} {\overset{\frac{\pi}{\mathrm{2}}} {\mid}}\:=\:−\mathrm{7}\left(−\:\mathrm{cos}\:{x}\right)\:+\:\mathrm{3}\left(\mathrm{sin}\:{x}\right)\:\underset{\frac{\pi}{\mathrm{4}}} {\overset{\frac{\pi}{\mathrm{2}}} {\mid}} \\ $$$$=\:\mathrm{7}\:\mathrm{cos}\:{x}\:+\:\mathrm{3sin}\:{x}\:\underset{\frac{\pi}{\mathrm{4}}} {\overset{\frac{\pi}{\mathrm{2}}} {\mid}} \\ $$$$ \\ $$$${Evaluate}\:{it}\:{to}\:{the}\:{top}\:{and}\:{bottom}\:{limit}\:{of}\:{integration}: \\ $$$$ \\ $$$$=\:\left(\mathrm{7}\:\mathrm{cos}\:\centerdot\:\frac{\pi}{\mathrm{2}}\:+\:\mathrm{3}\:\mathrm{sin}\:\centerdot\:\frac{\pi}{\mathrm{2}}\right)−\:\left(\mathrm{7}\:\mathrm{cos}\:\centerdot\:\frac{\pi}{\mathrm{4}\:}\:\:+\:\mathrm{3}\:\mathrm{sin}\:\centerdot\:\frac{\pi}{\mathrm{4}}\:\right) \\ $$$$=\left[\mathrm{7}\left(\mathrm{0}\right)\:+\:\mathrm{3}\left(\mathrm{1}\right)\right]\:−\:\left[\mathrm{7}\left(\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\right)\:+\:\mathrm{3}\left(\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\right)\right] \\ $$$$=\:\mathrm{3}\:−\:\frac{\mathrm{7}\sqrt{\mathrm{2}}}{\mathrm{2}}\:−\:\frac{\mathrm{3}\sqrt{\mathrm{2}}}{\mathrm{2}} \\ $$$$=\:\mathrm{3}\:−\:\frac{\mathrm{10}\sqrt{\mathrm{2}}}{\mathrm{2}}\:{or}\:\mathrm{3}\:−\:\mathrm{5}\sqrt{\mathrm{2}} \\ $$$$ \\ $$$${Answer}:\:\mathrm{3}\:−\:\mathrm{5}\sqrt{\mathrm{2}} \\ $$$$ \\ $$$$\left.{Solution}\:{by}\:{Roswel}:\right) \\ $$

Question Number 150034 Answers: 1 Comments: 0

Find a closed form: a∈R and a≠0 Ω(a)=∫_( 0) ^( ∞) (x^4 /((1+x^2 )(1+a^4 x^4 ))) dx

$$\mathrm{Find}\:\boldsymbol{\mathrm{a}}\:\mathrm{closed}\:\mathrm{form}:\:\:\boldsymbol{\mathrm{a}}\in\mathbb{R}\:\:\mathrm{and}\:\:\boldsymbol{\mathrm{a}}\neq\mathrm{0} \\ $$$$\Omega\left(\mathrm{a}\right)=\underset{\:\mathrm{0}} {\overset{\:\infty} {\int}}\frac{\mathrm{x}^{\mathrm{4}} }{\left(\mathrm{1}+\mathrm{x}^{\mathrm{2}} \right)\left(\mathrm{1}+\mathrm{a}^{\mathrm{4}} \mathrm{x}^{\mathrm{4}} \right)}\:\mathrm{dx} \\ $$

Question Number 150033 Answers: 1 Comments: 0

Question Number 150030 Answers: 0 Comments: 0

Roll a fair die twice and define A to be event that the sum of the scores showing up is greater than 7, B be the event that the sum of the scores showing up is a multiple of 3 and C be the event that the sum of the scores showing up is a prime number. Which of the events A,B and C are independent event? are the 3 events jointly independent?

Question Number 150029 Answers: 1 Comments: 0

Calcular: (√(8 + 2(√(8 + 2(√(8 + ...)))))) = ?

$$\mathrm{Calcular}: \\ $$$$\sqrt{\mathrm{8}\:+\:\mathrm{2}\sqrt{\mathrm{8}\:+\:\mathrm{2}\sqrt{\mathrm{8}\:+\:...}}}\:=\:? \\ $$

Question Number 150017 Answers: 0 Comments: 2