If ∫ tf(t)dt ∈ to (0, x)=x^2f(x) and f(2)=3, then f(6) equal to (1) 1 (2) 6 (3) 3 (4) 2

Question

If \(\int_0^x \text{tf}(t) d t=x^2 f(x)\text{ and }f(2)=3,\) then f(6) equals to

(1) 1

(2) 6

(3) 3

(4) 2

Answer 1

Correct option is (1) 1  

\(\int_0^x \text{tf}(t) d t=x^2 f(x)\)   

Differentiating both sides w.r.t 'x'  

\(x f(x)=x^2 f(x)+2 x f(x) \)

\(\frac{x^2 d y}{d x}+x y=0 \)

\(\frac{d y}{y}=\frac{-d x}{x} \)

\(\ln y+\ln x=\ln c \)

\(y x=c \)

\(\text{As}\ f(2)=3 \)

6 = c 

\(\therefore y x=6 \)

\(\therefore \text { Put } x=6 \)

y(6) = 6 

y = 1