The number of values of k for which the equation x2–2x+k=0 has two distinct roots lying in the interval (0, 1) is

Let the two distinct roots lying between 0 and 1 be α,β such that α<β.As f(x)=x2−2x+k is a differentiable and  f(α)=f(β)=0, therefore by the Roll’s theorem there exist γ∈(α,β)⊂(0,1) such that f′(γ)=0⇒2γ−2=0⇒γ=1.Not possible.