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f(x)=x / logx find interval in which it decrease or increase?

f(x) =xlog xf'(x)=log x-1(logx)2Now the function is increasing or decreasing in a particular intervals depends upon the sign of derivatives The function is increasing in the interval where the derivative is positive and decreasing in the intervals where the derivative is negativeHence for increasing in a particular interval  f'(x) >0⇒log x-1(logx)2>0⇒log x >1⇒x>e And for decreasing in a particular interval, f'(x) <0⇒log x-1(logx)2<0⇒log x <1⇒x<e   but the domain of f(x) is x>0 Hence f(x) is increasing in (e,∞)   and decreasing in(0,e) and at x=e there is a minima

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You can observe the graph and say that it is strictly decreasing in the interval (0,infinity)..

Or otherwise just differentiate the given function and assign it to '0' and substitute the obtained values, you will get all values positive for x<e and x>e.. so the given function is either strictly increasing or strictly decreasing.

To decide whether increasing or decreasing,compare any two values at different x positions.. for xample when x=e,value =e and when x=1,value is +infinity.. since 1<e,if u draw a graph u wil observe that it is strictly decreasing in that interval..  similarly the other intervals

 

Hope u understood :)

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