Find the interval on which the function x/logx is increasing or decreasing.
Manvendra Singh chahar , 10 Years ago
Grade Upto college level
Algebra> Find the interval on which the function x...
1 Answers
SHAIK AASIF AHAMED
Last Activity: 10 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
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