The minimum area bounded by the function y=f(x) and y= x+9( R) where f satisfies the relation f(x+y)=f(x)+f(y)+y f(x) x, y R and f(0)=0 is 9 A . value of A is

Step-by-step solution

aligned & f(x)= _ h 0 f(x+h)-f(x+0) h _ h 0 f(x)+f(h)+h f(x) -f(x)-f(0)-0 f(x) h & _ h 0 ( f(h)-f(0) h-0 )+ f(x) & f(x)= f(x) & f^ (x) f(x) d x= d x & 2 f(x) =x+c & f(x)= x^2 4 & When =0 arca is minimum & Required minimum area =2 ₀^2 2 y d y 4 ( y^ 3 / 2 3 / 2 )₀^9=72 sq. unit aligned