Let f : R → R   a n d   g : R → R be two non-constant differentiable functions. If f…

Let f:R→R and g:R→R be two non-constant differentiable functions. If f′x=efx-gxg′x for all x∈R, and f1=g2=1 then which of the following statement(s) is (are) TRUE?
  1. f2<1-loge⁡2
  2. f2>1-loge⁡2
  3. g1>1-loge⁡2
  4. g1<1-loge⁡2

Solution

f′x=efx-gxg′x∀x∈R
⇒  e-fx.f′x-e-gxg′x=0
⇒  ∫e-fxf′x-e-gx.g′xdx=C
⇒ -e-fx+e-gx=C
⇒ -e-f1+e-g1=-e-f2+e-g2
⇒ -1e+eg1=-e-f2+1e
⇒  e-f2+e-g1=2e
∴  e-f2<2eand e-g1<2e
⇒ -f2<ln⁡2-1 and-g1<ln⁡2-1
⇒  f2>1-ln⁡2 and g1>1-ln⁡2

Asked in: JEE Advanced 2018 (Paper 1)

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