Question

JEE ADVANCED MATHS QUESTION SEPTEMBER 2020

Answer 1

Let b be a non zero real number. suppose f : R → R is a differentiable function such that f(0) = 1. if the derivative f' of f satisfies the equation,

f'(x) = f(x)/(b² + x²)

for all x ∈ R, then which of the following statements is/are TRUE ?

solution : here f'(x) = f(x)/(b² + x²)

⇒∫f'(x)/f(x) dx = ∫ dx/(b² + x²)

⇒ln|f(x)| = 1/b tan¯¹(x/b) + C

here given f(0) = 1

so, ln|f(0)| = 1/b tan¯¹(0/b) + C

⇒c = 0

so, ln|f(x)| = 1/b tan¯¹ (x/b)

|f(x)| = e^{1/b tan¯¹ (x/b) }

f(x) = ± e^{1/b tan¯¹ (x/b) }

since f(0) = 1, f(x) = e^{1/b tan¯¹ (x/b) }

f(x) is increasing for all real value of x.

so, if b > 0, then f is increasing function, option (A) is correct.

option (B) is incorrect.

when x tends to -x ,

f(-x) = e^{1/b tan¯¹ (-x/b) } = e^{-1/b tan¯¹ (x/b) }

f(x).f(-x) = e^{1/b tan¯¹ (x/b) }.e^{-1/b tan¯¹ (x/b) } = e⁰ = 1

option (C) is also correct.

as you can see option (D) is incorrect

Therefore options (A) and (C) are correct choices.