Question

JEE Mains Maths question is in the image

Answer 1

Answer:

(4) increasing in (-1,0) and decreasing in ( 0,infinity )

Answer 2

It has given that f(x) = ln(1 + x)/x , x ∈ (-1, ∞) and f(0) = 1

we have to check f(x) is ...

1. decreasing in (-1, 0) and increasing in (0, ∞)
2. always increasing
3. always decreasing
4. increasing in (-1, 0) and decreasing in (0, ∞)

solution : function, f(x) = ln(1 + x)/x

differentiating f(x) w.r.t x we get,

f'(x) = {xd(ln(1 + x))/dx - ln(1 + x) dx/dx}/x²

= {x/(1 + x) - ln(1 + x)}/x²

= {x - (1 + x)ln(1 + x)}/(1 + x)x²

in (-1, ∞), (1 + x)x² > 0

so, we have to check {x - (1 + x)ln(1 + x) }

let g(x) = x - (1 + x)ln(1 + x)

but g(x) < 0 for all x ∈ (-1, ∞)

so, f'(x) < 0 for all x ∈ (-1, ∞)

Therefore f(x) is always decreasing in (-1,∞) i.e., the correct option is (3).