1. The problem statement, all variables and given/known data Let f(x) be a non-constant twice differentiable function defined on R such that f(x)=f(1-x) and f'(1/4) =0 then what is the minimum number of real roots of the equation (f"(x))^2+f'(x)f'''(x)=0. 3. The attempt at a solution f'(x)=-f'(1-x) f"(x)=f"(1-x) f'''(x)=-f'''(1-x) putting x = 1/4 in the first eqn gives f'(3/4)=0 putting x=1/4 in the given equation (f"(x))^2+f'(x)f'''(x)=0 gives f"(1/4)=0. ∴f"(3/4)=0. So I have got a total of 2 roots. But there are some more which I can't find.