For every twice differentiable function f:R→[-2,2] with (f(0))²+(f'(0))²=85, which of the following statement(s) is (are) TRUE?

The correct options are
A. There exist r, s∈R, where r<s, such that f is one-one on the open interval (r, s)
B. There exists x₀∈(-∞,0) such that |f'(x₀)|≤1
D. There exists α∈(-∞,4) such that f(α)+f''(α)=0 and f'(α)≠0
f(x) can't be constant throughout the domain. Hence we can find x∈(r, s) such that f(x) is one-one. Hence option (A) is true.
Option (B): |f'(x₀)| = ||f(0)-f(-∞)/4|| ≤ 1 (LMVT)
Option (C): f(x)=sin(√85x) satisfies given condition but limx→∞sin(√85x) D.N.E. ⇒ Incorrect
Option (D): g(x) = f²(x)+(f'(x))²
|f'(x₁)|≤1 (by LMVT)
|f(x₁)|≤2 (given) ⇒ g(x₁)≤5 ∃ x₁∈(-∞,0)
Similarly, g(x₂)≤5 ∃ x₂∈(0,4)
g(0)=85 ⇒ g(x) has maxima in (x₁,x₂), say at α
g'(α)=0 g(α)≥85
2f'(α)(f(α)+f''(α))=0
If f'(α)=0 ⇒ g(α)=f²(α)≥85
Not possible ⇒ f(α)+f''(α)=0 ∃ α ∈(x₁,x₂)∈(-∞,4)
Option (D) correct.