Let f : R → R be a twice differentiable function such that f ( x + y ) = f ( x ) f ( y ) for all x , y ∈ R . If f ′ ( 0 ) = 4 a and f satisfies f ′′ ( x ) − 3 a f ′ ( x ) − f ( x ) = 0 , a 0 , then

Question

Let f : R → R be a twice differentiable function such that f ( x + y ) = f ( x ) f ( y ) for all x , y ∈ R . If f ′ ( 0 ) = 4 a and f satisfies f ′′ ( x ) − 3 a f ′ ( x ) − f ( x ) = 0 , a > 0 , then the area of the region R = {( x , y ) ∣ 0 ≤ y ≤ f ( a x ) , 0 ≤ x ≤ 2 } is:

TestHub · Accepted Answer

​ f ( x + y ) = f ( x ) ⋅ f ( y ) ⇒ f ( x ) = e λ x f ′ ( 0 ) = 4 a ⇒ f ′ ( x ) = λ e λ x ⇒ λ = 4 a ​ So, f ( x ) = e 4 x ​ f ′′ ( x ) − 3 af ′ ( x ) − f ( x ) = 0 ⇒ λ 2 − 3 a λ − 1 = 0 ⇒ 16 a 2 − 12 a 2 − 1 = 0 ⇒ 4 a 2 = 1 ⇒ a = 2 1 ​ ​ ​ F ( x ) = e 2 x Area = ∫ 0 2 ​ e x dx = e 2 − 1 ​

Mathematics - Area Under Curves Question with Solution | TestHub

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