JEE Main20202 Sep 2020Evening ShiftMathematicsApplication of DerivativesActual
Let f : - 1 , ∞ → R be defined by f 0 = 1 and f x = 1 x log e 1 + x , x ≠ 0 . Then the function f
Options
- ADecreases in - 1 , 0 and increases in 0 , ∞
- BIncreases in - 1 , ∞
- CIncreases in - 1 , 0 and decreases in 0 , ∞
- DDecreases in - 1 , ∞
Correct answer
D. Decreases in - 1 , ∞
Step-by-step solution
f x = 1 x log e 1 + x ,   x ≠ 0 f ' x = x 1 + x - ln 1 + x x 2 = x - 1 + x ln 1 + x x 2 1 + x Let g x = x - 1 + x ln 1 + x g ' x = - ln 1 + x ⇒ g ' x > 0 ∀ x ∈ - 1 , 0 < 0 ∀ x ∈ 0 , ∞ g m a x at x = 0 ⇒ g 0 = 0 is the maximum value So, g x < 0 ∀ x ∈ - 1 , ∞    ⇒   f ' x < 0 ∀ x ∈ - 1 , ∞ Hence, f x is decreasing ∀ x ∈ - 1 , ∞