Question

Find a polynomial $f\left(x\right)$ such that $f\left(2x\right)=f^{\prime }\left(x\right)f^{\prime \prime }\left(x\right)$.

Answer 1

Since no extra condition is imposed, let us take $f\left(x\right)=e^x$

According to equation,

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

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

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

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

Thus, $f\left(x\right)=e^x$ satisfies the condition.