Let f: R R be a twice differentiable function such that f(x+y)=f(x) f(y) for all x, y R and f(x)>0 . The area of the region bounded by the curve y=f(x) , the x-axis, the y-axis, and the line x=1 is e^3-1 3 . If f(x) also satisfies the differential equation f^ (x)- f^ (x)+15 f(x)=0 , then the value of is:

Step-by-step solution

f(x+y)=f(x) f(y) and f(x)>0 implies f(x)=e^ k x for some constant k . The area of the region bounded by y=f(x) , x=0 , x=1 , and y=0 is given by: ₀^1 e^ k x d x= [ e^ k x k ]₀^1= e^k-1 k Given that the area is e^3-1 3 , we get k=3 . So, f(x)=e^ 3 x . Now, find the derivatives: f^ (x)=3 e^ 3 x f^ (x)=9 e^ 3 x Substitute these into the given differential equation f^ (x)- f^ (x)+15 f(x)=0 : 9 e^ 3 x - (3 e^ 3 x )+15 e^ 3 x =0 24-3 =0 =8 Answer: 8