x*(1/x)= 1, not 0!

The derivative of x ln(x) is 1+ ln(x), not ln(x), but the second derivativeAccording to lnx graph, f(x) decreasing when 0 < x < 1

b) f''(x) = 1 / x

According to 1 / x graph f concave upward when x < 0is1/x so this is correct.

For what values of x is 1+ ln(x)< 0?