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Question Number 208594 by Jimenez000 last updated on 18/Jun/24

Find  x+3^x <4

Findx+3x<4

Answered by mr W last updated on 18/Jun/24

3^x <4−x  (4−x)3^(−x) >1  (4−x)3^(4−x) >3^4   (4−x)ln 3 e^((4−x)ln 3) >3^4 ln 3  (4−x)ln 3>W(3^4 ln 3)  x<4−((W(3^4 ln 3))/(ln 3))=1

3x<4x(4x)3x>1(4x)34x>34(4x)ln3e(4x)ln3>34ln3(4x)ln3>W(34ln3)x<4W(34ln3)ln3=1

Commented by Jimenez000 last updated on 18/Jun/24

From step 3 to 4, what exactly did do you?, i can′t see it

Fromstep3to4,whatexactlydiddoyou?,icantseeit

Commented by mr W last updated on 18/Jun/24

say W(3^4 ln 3)=a  ae^a =3^4 ln 3=3^3 (3ln 3)=(3 ln 3)e^((3 ln 3))   ⇒a=3 ln 3  4−((W(3^4 ln 3))/(ln 3))=4−((3 ln 3)/(ln 3))=4−3=1

sayW(34ln3)=aaea=34ln3=33(3ln3)=(3ln3)e(3ln3)a=3ln34W(34ln3)ln3=43ln3ln3=43=1

Commented by Jimenez000 last updated on 18/Jun/24

Thank you so much mr W

ThankyousomuchmrW

Commented by Kademi last updated on 18/Jun/24

    W(xe^x ) = x   W(a^c ln a) = W(a^(a^b +b) ln a) = W(a^b ln a∙e^(a^b ln a) ) = a^b ln a

W(xex)=xW(aclna)=W(aab+blna)=W(ablnaeablna)=ablna

Answered by Jimenez000 last updated on 18/Jun/24

3^x <4−x  xln 3<ln (4−x)  x<((ln (4−x))/(ln 3))  I don′t know how to continue

3x<4xxln3<ln(4x)x<ln(4x)ln3Idontknowhowtocontinue

Commented by Kademi last updated on 18/Jun/24

 x < ((ln (4−x))/(ln 3))   x < ((4−x)/3)  3x < 4−x  4x−4 < 0   4(x−1) < 0    { ((4 < 0 ⇒ 4 > 0)),((x−1 < 0 ⇒ x < 1)) :}

x<ln(4x)ln3x<4x33x<4x4x4<04(x1)<0{4<04>0x1<0x<1

Answered by mr W last updated on 18/Jun/24

f(x)=x+3^x   f′(x)=1+3^x ln 3>0 ⇒strictly increasing  f(1)=1+3^1 =4  f(x)=x+3^x <4=f(1)  ⇒x<1

f(x)=x+3xf(x)=1+3xln3>0strictlyincreasingf(1)=1+31=4f(x)=x+3x<4=f(1)x<1

Answered by Kademi last updated on 18/Jun/24

    3^x  < 4−x   e^(xln 3)  < 4−x   1 < (4−x)e^(−xln 3)    ln 3e^(4ln 3)  < (4−x)ln 3e^(4ln 3−xln 3)    ln 3e^(ln 3^4 )  < (4−x)ln 3e^((4−x)ln 3)    W(3^4 ln 3) < W((4−x)ln 3e^((4−x)ln 3) )   W(3^4 ln 3) < (4−x)ln 3   ((W(3^4 ln 3))/(ln 3)) < 4−x   x < 4−((W(3^4 ln 3))/(ln 3))   x < 4−((W(3ln 3e^(3ln 3) ))/(ln 3))   x < 4−((3ln 3)/(ln 3))   x < 4−3   x < 1

3x<4xexln3<4x1<(4x)exln3ln3e4ln3<(4x)ln3e4ln3xln3ln3eln34<(4x)ln3e(4x)ln3W(34ln3)<W((4x)ln3e(4x)ln3)W(34ln3)<(4x)ln3W(34ln3)ln3<4xx<4W(34ln3)ln3x<4W(3ln3e3ln3)ln3x<43ln3ln3x<43x<1

Commented by Jimenez000 last updated on 18/Jun/24

Thank you Kademi

ThankyouKademi

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