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Find the maximum value of f(x) =( 1/x)^x

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@aaditya 

We have our functionf(x)=(1/x)^x

Or we’ll be using the equation,

y=(1/x)^x

Taking ln both side we get

lny=−xlnx 

Differentiating both sides with respect to x,

(dy/dx)/y=−lnx−1 

dy/dx=−y(lnx+1) 

Equating dy/dx  to 0, we get

−y(lnx+1)=0 

Since y is an exponential function it can never be equal to zero, hence:

lnx+1=0 

lnx=−1 

x=e−1 

So, for the maximum value we put x=e−1 in f(x)

 to get the value of f(x)  at the point.

f(e^{-1})=e^{1/e}

Hence the maximum value of the function is e^{1/e}

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avinash.dongre

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