Let b be a nonzero real number. Suppose f : ℝ → ℝ is a differentiable function such that f…

Let b be a nonzero real number. Suppose f:ℝ→ℝ is a differentiable function such that f0=1. If the derivative f' of f satisfies the equation f'x=fxb2+x2 for all x∈ℝ, then which of the following statements is/are TRUE?
  1. If b>0, then f is an increasing function
  2. If b<0, then f is a decreasing function
  3. fx f-x=1 for all x∈ℝ
  4. fx-f-x=0 for all x∈ℝ

Solution

f'x=fxb2+x2⇒f'xfx=1b2+x2

⇒lnfx=1btan-1xb+C

∴f0=1    ∴0=0+C⇒C=0

∴ lnfx=1btan-1xb

⇒fx=e1btan-1xb

∴f'x=e1btan-1xb.1b2+x2>0 for x∈R

⇒fx is increasing for x∈R.

fx.f-x=e1btan-1xb·e-1btan-1xb=e0=1.

Asked in: JEE Advanced 2020 (Paper 2)

Practice more Differential Equations questions on Aicharya