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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 (B) There exists x0  (–4, 0) such that |f(x0)|  1 (C) x lim f(x) 1   (D) There exists   (–4, 4) such that f() + f() = 0 and f()  0

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
(B) There exists x0  (–4, 0) such that |f(x0)|  1
(C) 
x
lim f(x) 1


(D) There exists   (–4, 4) such that f() + f() = 0 and f()  0

Grade:12

1 Answers

Arun
25757 Points
4 years ago

L.M.V.T. in [–4, 0]

|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

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