Question : Show that $\displaystyle \frac{logx}{x}$ has a maximum value at $\displaystyle x=e$. Compute the maximum value
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Solution:
x=e is the confusion what to do with it
He said solve for x! You don't "put in" any value of x.
(Although, here, since you are given a value of x, x= e, you could put that in and see if the derivative is 0 there. You might also want to check that the second derivative is negative there (second derivative test) or that the first derivative is positive for x< e and negative for x> e (first derivative test).