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:

Step-by-step solution

aligned & f(x+y)=f(x) f(y) & f(x)=e^ x f^ (0)=4 a & f^ (x)= e^ x =4 a aligned So, f(x)=e^ 4 x aligned & f ^ ( x )-3 af ^ ( x )- f ( x )=0 & ^2-3 a -1=0 & 16 a ^2-12 a ^2-1=0 4 a ^2=1 a = 1 2 aligned aligned & F ( x )= e ^ 2 x & Area = ₀^2 e ^ x dx = e ^2-1 aligned