Let f : - 1 , ∞ → R be defined by f 0 = 1 and f x = 1 x log e 1 + x ,   x ≠ 0 . Then…

Let f:-1,∞→R be defined by f0=1 and fx=1xloge1+x, x≠0. Then the function f
  1. Decreases in-1,0 and increases in 0,∞
  2. Increases in -1,∞
  3. Increases in -1,0 and decreases in 0,∞ 
  4. Decreases in -1,∞

Solution

fx=1xloge1+x, x≠0

f'x=x1+x-ln1+xx2=x-1+xln1+xx21+x

Let gx=x-1+xln1+x

g'x=-ln1+x

⇒g'x>0∀x∈-1,0<0∀x∈0,∞

gmax at x=0⇒g0=0 is the maximum value

So, gx<0∀x∈-1,∞  ⇒ f'x<0∀x∈-1,∞

Hence, fx is decreasing ∀x∈-1,∞

Asked in: JEE Main 2020 (02 Sep Shift 2)

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