Question

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? 

(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 ∈ (-4, 0) such that |f'(x₀)| ≤ 1 

(C)  lim_(x→∞)f(x) = 1

(D) There exist α ∈ (-4, 4) such that f(α) + f''(α) = 0 and f'(α) ≠ 0

Answer 1

solution A,B,D

L.M.V.T. in [–4, 0]

Similarly |f'(x₁)| ≤ 1 for some x₁ ∈ (0, 4)
g(x) = (f(x))² + (f'(x))²
g(x₀) ≤   5, g(x₁) ≤   5
g(0) = 85 it has a local maximum having value ≥ 85
Say α
g'(α) = 0, g''(α) ≤ 0
2f(α)f'(α) + 2f'(α)f''(α) = 0
f'(α)(f(α) + f'(α)) = 0
as f'(α) ≠ 0