For every twice differentiable function f:ℝ→−2,2 with f02+f102=85 which of the Following statement(s) is (are) TRUE?

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There exist r,s∈ℝ, where r

There exists x0∈−4,0 such that f1x0≤1

limx→∞fx=1

There exists α∈−4,4 such that fα+f11α=0 and f1α≠0

answer is A.

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Detailed Solution

ƒ(x) can't be constant throughout the domain. Hence we can find x (r, s) such that ƒ(x) is one-one ∴Option (A) is truef1x0=f0−f−44≤1∴Option (B) is truefx=sin85x satisfies the given condition but limx→∞sin85x but does not exist.∴Option (C) is not true. Let gx=f2x+f1x2Since f1x1≤1 and fx1≤2 it gives ⇒gx1≤5 for some x1∈−4,0g(0)=85⇒g(x) has maxima in x1,x2 say at α, g1(α)=0.⇒ 2f1(α)f(α)+f11(α)=0 If f1(α)=0⇒g(α)=f2(α)=85 not possible ⇒f(α)+f11α=0 for some α∈x1,x2∈(-4,4)∴Option (D) is correct

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