Let f be a twice differentiable function defined on

Given $f(x) f^{\prime \prime}(X)-\left(f^{\prime}(x)\right)^{2}=0$

Let $h(x)=\frac{f(x)}{f^{\prime}(x)}$

$\Rightarrow h^{\prime}(x)=0 \quad \Rightarrow h(x)=k$

$\Rightarrow \frac{f(x)}{f^{\prime}(x)}=k \quad \Rightarrow f(x)=k f^{\prime}(x)$

$\Rightarrow f(0)=k f^{\prime}(0) \quad \Rightarrow 1=k(2) \Rightarrow k=\frac{1}{2}$

Now $f(x)=\frac{1}{2} f^{\prime}(x) \Rightarrow \int 2 d x=\int \frac{f^{\prime}(x)}{f(x)} d x$

$\Rightarrow 2 x=\ln |f(x)|+C$

As $f(0)=1 \Rightarrow C=0$

$\Rightarrow 2 x=\ln |f(X)| \Rightarrow f(x)=\pm e^{2 x}$

As $f(0)=1 \Rightarrow f(x)=e^{2 x} \Rightarrow f(1)=e^{2}$