For every twice differentiable function f : R → - 2 ,   2   w i t h   f 0 2 + f ′…

For every twice differentiable function f:R→-2, 2 with f02+f′02=85 , which of the following statement(s) is (are) TRUE?
  1. There exist r,s∈R, where r<s,  such that f is one-one on the open interval (r,s)
  2. There exists x0∈-4,0 such that f′x0≤1
  3. limx→∞⁡fx=1
  4. There exists α∈-4, 4 such that fα+f′′α=0 and f′α ≠0

Solution

fx can't be constant throughout the domain. Hence we can find x∈r,s such that ƒ(x) is one-one option (A) is true.
(B) f′x=f0-f-44≤1 LMVT
(C) fx=sin⁡85x satisfies given condition
But limx→∞⁡sin⁡85 D.N.E
⇒   Incorrect
D gx=f2x+f′x2
f′x1≤1 (by LMVT)
fx1≤2 (given)
⇒  gx1≤5  ∃x1∈-4,0
Similarly gx2≤5  ∃x2∈0,4
g0=85 ⇒gx has maxima in x1,x2 say at α
g′α=0 and gα≥85
2f′αfα+f′′α=0
If f′α=0⇒gα=f2α≥85 not possible
⇒  fα+f′′α=0 ∃α∈x1,x2∈-4,4
(D) correct

Asked in: JEE Advanced 2018 (Paper 1)

Practice more Hyperbola questions on Aicharya