Question

\( \int_{0}^{x} t f(t) d t=x^{2} f(x) \) and \( f(2)=3 \), then find the value of \( f(6) \).

Answer 1

Applying derivative on both sides we get

xf(x)=x²f^I(x)+2xf(x) 

-xf(x)=x²f^I(x)

\(f^I(x)/f(x)=-1/x\)

Applying integral on both sides we get

f(x)=c/x where c is an constant

given f(x) = 3 when x is 2 

so 3=c/2 so c is 6 

therefore function is f(x)=x/6

so f(6) is 1