Limits, Continuity and Differentiability Part 2

f(0)=f(1)=f′(0)=0
Apply Rolles theorem on y=f(x) in x∈[0,1]
f(0)=f(1)=0
⇒f′(α)=0 where α∈(0,1)
Now apply Rolles theorem on y=f′(x) in x∈[0,α]
f′(0)=f′(α)=0 and f′(x) is continuous and differentiable
f′(0)=f′(α)=0 and f′(x) is continuous and differentiable
⇒f"(β)=0 for some ,β∈(0,α)∈(0,1)
⇒f"(x)=0 for some x‌in∈(0,1)