The number of values of a  for which the equation x3−3x+a=0  has two distinct real roots lying in the interval (0,1)  are :

f(x)=x3+3x+a, x∈[0,1] Let f(x)=0  has two distinct roots α,β in (0,1)  0<α<β<1 . Then by Rolle’s theorem f(x)=3x2−3=0  at least one in (α,β) . But f'(x)=0  has two roots –1 and 1.Hence x3−3x+a=0  cannot have two distinct roots in (0,1)  for any a∈R .