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x-2-4x-3-x-2-6x-4-1-

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Question Number 133934 by liberty last updated on 25/Feb/21
(x^2 −4x+3)^(x^2 −6x+4) ≤ 1
(x2−4x+3)x2−6x+4⩽1
Answered by EDWIN88 last updated on 25/Feb/21
(i) x^2 −4x+3 > 0 ; { ((x<1)),((x>3)) :} (ii) ln (x^2 −4x+3)^(x^2 −6x+4) ≤ ln 1 ⇒(x^2 −6x+4)ln (x^2 −4x+3) ≤ 0 { ((x^2 −6x+4=0⇒x_1 =3+(√5) ; x_2 =3−(√5))),((ln (x^2 −4x+3) =0⇒x^2 −4x+3=1 ; x_1 =2+(√2) ;x_2 =2−(√2))) :} ___+ − + + − − ___ 2−(√2) 3−(√5) 1 3 2+(√2) 3+(√5) solution set is [ 2−(√2) ; 3−(√5) ] ∪ [ 2+(√2) ; 3+(√5) ]
(i)x2−4x+3>0;{x<1x>3(ii)ln(x2−4x+3)x2−6x+4⩽ln1⇒(x2−6x+4)ln(x2−4x+3)⩽0{x2−6x+4=0⇒x1=3+5;x2=3−5ln(x2−4x+3)=0⇒x2−4x+3=1;x1=2+2;x2=2−2___+−++−−___2−23−5132+23+5solutionsetis[2−2;3−5]∪[2+2;3+5]
Answered by bobhans last updated on 25/Feb/21
⇔ (x^2 −4x+3)^(x^2 −6x+4) ≤ (x^2 −4x+3)^0 (x^2 −4x+3−1)(x^2 −6x+4−0) ≤ 0 (x^2 −4x+2)(x^2 −6x+4) ≤ 0 {(x−2)^2 −2}{(x−3)^2 −5} ≤ 0 (x−2+(√2))(x−2−(√2))(x−3+(√5))(x−3−(√5) ) ≤ 0 (x−(2−(√2) ))(x−(2+(√2) ))((x−(3−(√5)))(x−(3+(√5))) ≤ 0 ⇒x ∈ [ 2−(√2) , 3−(√5) ] ∪ [ 2+(√2) ,3+(√5) ]
⇔(x2−4x+3)x2−6x+4⩽(x2−4x+3)0(x2−4x+3−1)(x2−6x+4−0)⩽0(x2−4x+2)(x2−6x+4)⩽0{(x−2)2−2}{(x−3)2−5}⩽0(x−2+2)(x−2−2)(x−3+5)(x−3−5)⩽0(x−(2−2))(x−(2+2))((x−(3−5))(x−(3+5))⩽0⇒x∈[2−2,3−5]∪[2+2,3+5]

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