MCQ
Let $f:[1, \infty) \rightarrow[2, \infty)$ be a differentiable function such that $f(1)=2$. If $6 \int_1^x f(t) d t=3 x f(x)-x^3$ for all $x \geq 1$, then the value of $f(2)$ is
- ✓$6$
- B$3$
- C$0$
- D$1$
Differentiating w.r.t. $x$, we get (Use Newton Leibnitz theorem for differentiating a definite integral)
$\Rightarrow 6 f(x)=3 x f^{\prime}(x)+3 f(x)-3 x^2 $
$\Rightarrow x f^{\prime}(x)=f(x)+x^2 $
$\text { or, } \frac{d y}{d x}-\frac{y}{x}=x$
This is a first order linear differential equation with Integrating factor
$e ^{-\int \frac{1}{ x } dx }= e ^{-\ln x }=\frac{1}{ x }$
Its general solution is $\frac{y}{x}=\int x \times \frac{1}{x} d x+C$
$\Rightarrow \frac{y}{x}=x+C $
$\text { Since } f(1)=2 $
$\Rightarrow C=1$
So $y=x^2+x $
$\Rightarrow f(2)=6$
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.