Let (f:(0, ) R ) be a twice differentiable function. If for some ( a 0, ₀^1 f( x) d = a f(x), f(1)=1 ) and (f(16)= 1 8 ), then (16-f^ ( 1 16 ) ) is equal to _______.

Step-by-step solution

Given, ₀^1 f( x) d =a f(x) Let x=ud = 1 x d u From (1) 1 x ₀^x f(u) d u=a f(x) ₀^x f(u) d u= axf (x) Differentiate both sides aligned & f(x)=a (x f^ (x)+f(x) ) & f(x)=a x f^ (x)+a f(x) & (1-a) f(x)=a x f^ (x) & f^ (x) f(x) = (1-a) a 1 x aligned Integrate both side w.r.t. ( x ) aligned & f^ (x) f(x) d x= (1-a) a 1 x d x & f(x)= ( 1-a a ) x+c aligned Now at x=1 f(1)=1 c=0 Also given f(16)= 1 8 aligned & 1 8 =(16)^ 1-a a & 2⁻³=2^ 4-4 a a & -3= 4-4 a a & -3 a=4-4 a & a=4 & f(x)=x^ -3 / 4 & f(x)= -3 4 x^ - 7 4 aligned a