Question

Let $f\left(x\right)$ be a differentiable function satisfying the equation $\frac{f^{\prime }\left(x\right)}{2}=\frac{x}{e^{f\left(x\right)}}\forall \in R$. If $f\left(1\right)=1$, then the number of solutions of the equation $f\left(x\right)=f\left(x\right)$ is-

• A. $1$
• B. $2$
• C. $3$
• D. $Noneofthese$

Answer 1

The correct option is D $Noneofthese$

Given that,

$\frac{f^{\prime }\left(x\right)}{2}=\frac{x}{e^{f\left(x\right)}}$

$⟹f^{\prime }\left(x\right).e^{f\left(x\right)}=2x$

$⟹\int f^{\prime }\left(x\right).e^{f\left(x\right)}dx=\int 2xdx$

$⟹e^{f\left(x\right)}=x^2+c$

$\because x=1,f\left(x\right)=f\left(1\right)=1$

$⟹e^{f\left(x\right)}=x^2+e-1$

$⟹f\left(x\right)={\log }_e(x^2+e-1)$

When, $f\left(x\right)=f\left(x\right)$ then the number of solutions are infinite.