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Find the interval on which the function x/logx is increasing or decreasing.

Find the interval on which the function x/logx is increasing or decreasing.

Grade:Upto college level

1 Answers

SHAIK AASIF AHAMED
askIITians Faculty 74 Points
7 years ago
f(x)=x/logx
f'(x)=logx−1/(logx)2
Nowthefunctionisincreasingordecreasinginaparticularintervalsdependsuponthesignofderivatives.
Thefunctionisincreasingintheintervalwherethederivativeispositiveanddecreasingintheintervalswherethederivativeisnegative.
Henceforincreasinginaparticularintervalf'(x)>0
⇒logx−1(logx)2>0
⇒logx>1
x>e
Andfordecreasinginaparticularinterval,f'(x)<0
⇒logx−1(logx)2<0
⇒logx<1
x<ebutthedomainoff(x)isx>0
Hencef(x)isincreasingin(e,∞)anddecreasingin(0,e)andatx=ethereisaminima
Thanks & Regards
Shaik Aasif Ahamed
askIITians Faculty

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