Question

If $f$ is a continuous function on $[0,2],$ differentiable in $(0,2)$ such that $f\left(2\right)=0,$ then which of the following options is(are) correct for atleast one value of $c\in (0,2)$

• A. $c^2{f}^{\prime }\left(c\right)+cf\left(c\right)=0$
• B. $c^2{f}^{\prime }\left(c\right)+2cf\left(c\right)=0$
• C. $c^3{f}^{\prime }\left(c\right)+c^{2}f\left(c\right)=0$
• D. $c^3{f}^{\prime }\left(c\right)+3c^{2}f\left(c\right)=0$

Answer 1

Answer: (B), (D)

$c^3{f}^{\prime }\left(c\right)+3c^{2}f\left(c\right)=0$

Let $g\left(x\right)=x^{n}f\left(x\right),n\in N$
$\left(1\right)g\left(x\right)$ is continuous on $[0,2]$
$(\because x^n\text{is continuous for all}x\in R,f(x\left)\text{is continuous on}\right[0,2\left]\right)$
$\left(2\right)g\left(x\right)$ is differentiable on $(0,2)$
$(\because x^n\text{is differentiable for all}x\in R,f(x\left)\text{is differentiable on}\right(0,2\left)\right)$
$\left(3\right)g\left(0\right)=g\left(2\right)=0$
All condition of rolle's theorem is satisfied.

$\therefore$ From rolle's theorem: $g^{\prime }\left(c\right)=0$ for atleast one value of $c\in (0,2)$
$\Rightarrow n{c}^{n-1}f\left(c\right)+c^n{f}^{\prime }\left(c\right)=0,n\in N$
$\Rightarrow 2cf\left(c\right)+c^2{f}^{\prime }\left(c\right)=0,n=2$
$\Rightarrow 3c^{2}f\left(c\right)+c^3{f}^{\prime }\left(c\right)=0,n=3$