Let f ( x ) be a differentiable function satisfying the equation 2 f ′ ( x ) = e f ( x ) x ∀ x ∈ R . If f ′ ( 1 ) = 1 , then the number of solution(s) of the equation f ′ ( x ) = f ( x ) is

Question

Let f ( x ) be a differentiable function satisfying the equation 2 f ′ ( x ) ​ = e f ( x ) x ​ ∀ x ∈ R . If f ′ ( 1 ) = 1 , then the number of solution(s) of the equation f ′ ( x ) = f ( x ) is

TestHub · Accepted Answer

∫ e f ( x ) ⋅ f ′ ( x ) dx = ∫ 2 xdx ⇒ e f ( x ) = x 2 + c ⇒ f ( x ) = ℓ n ( x 2 + c ) Now, f ′ ( x ) = x 2 + c 1 ​ ⋅ 2 x ⇒ f ′ ( 1 ) = 1 + c 2 ​ = 1 ⇒ c = 1 ⇒ f ( x ) = ℓ n ( x 2 + 1 ) Now, f ( x ) = f ′ ( x ) ⇒ ℓ n ( x 2 + 1 ) = x 2 + 1 2 x ​ Hence 2 solutions.

Mathematics - Indefinite Integration Question with Solution | TestHub

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JEE Advanced 2027 (Class 12)