Our purpose is to prove that F(x) take that same value at three different points,
In this way, it can be proved by Rolle theorem that F'(x) has two different zeros.
That is, f (x) f'' (x)+(f' (x)) 2 = 0 has two zeros.
Think about what's so special about this problem.
We know from the first question that f(η)=0, so it is easy to assume that F(x)=0.
F(η)=f(η) f'(η)=0，
Is there a one o'clock?
There is a trap in this problem, that is, f(0)=0, not.
The existence of limit must be finite, f (x) = f (x)/x x; Bounded × infinitesimal =0
(In fact, it is not difficult to understand that f(x) must be the same order infinitesimal of x, even if it is verified by finding f(x)=x, it is not wrong to get f (0).
So f (0) = f (0) f' (0) = 0.
The first two F(x)=0 have been used, and f'(x) has never been used. Generally speaking, the topic will make f'(x) also get 0.
Because f(0)=f(η), there are f' (ξ) = 0 and f (ξ) = 0,0.
Because F(0)= F(ξ)=0, F'(a)=0, and a∈(0, ξ) is contained in (0, 1).
Because F(ξ)= F(η)=0, F'(b)=0, and b∈(ξ, η) is contained in (0, 1).
That is, f (x) f'' (x)+(f' (x)) 2 = 0 has two zeros A and B ∈ (0, 1).