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.
The Attempt at a Solution
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.