We are asked to locate any relative extrema or inflection points for the graph of `y=xlnx ` :
The domain of the function is x>0.
Extrema can only occur at critical points; that is when the first derivative is zero or fails to exist.
`y'=lnx+x*1/x ==> y'=lnx + 1 `
This function is continuous and differentiable for all x in the domain, so setting y'=0 we get:
`lnx+1=0 ==> lnx=-1 ==> x=1/e~~0.368 `
For 0<x<1/e the first derivative is negative and for x>1/e it is positive, so the only extrema is a minimum at x=1/e.
Inflection points can only occur when the second derivative is zero:
`y''=1/x>0 forall x ` so there are no inflection points.