The correct option is D f(x) is decreasing on (0, 1)∪(1, e)
Given the function f(x) = xlog x⇒f'(x)=log x−1(log x)2For f(x) to be an increasing function, we must havef'(x)>0⇒log x−1(log x)2>0⇒log x−1>0 [∵(log x)2>0 for x>0]⇒log x>1⇒x>e⇒x∈(e, ∞)For f(x) to be a decreasing function, we must havef'(x)<0⇒log x−1(log x)2<0⇒log x−1<0 [∵(log x)2>0 for x>0]⇒log x<1⇒x<e⇒x∈(0, e) −{1} [∵f(x) is defined for x>0 and x≠1]So, f(x) is decreasing on (0, 1)∪(1, e).