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To find where a function is increasing and decreasing, you need to find where the derivative of the function is greater than zero or less than zero.

`f(x)=x^7lnx` differentiate using product rule

`f'(x)=7x^6lnx+x^7/x`

`=x^6(7lnx+1)`

Since `x^6` is always positive, we need to determine when `7lnx+1` is greater than zero or less than zero. We first find when it is equal to zero, then it will be positive on one side and negative on the other.

`7lnx+1=0`

`7lnx=-1`

`lnx=-1/7` switch to exponential form

`x=e^{-1/7}`

**This means that when `(e^{-1/7},infty)` , then the derivative is positive, which means the function is increasing, and when `(0,e^{-1/7})` , the derivative is negative, which means the function is decreasing.**

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