 Solution

At zero, the Fourier transform is $\hat{f}(0)=\int f=1$, so $\lambda f(0)=1$. In particular $\lambda \ne 0$ and $f(0)=1/\lambda$. The Fourier inversion theorem at zero also givesEquating the two values yields $\boxed{{\lambda }^2=2\pi }$. Since the explicit normalized Gaussian function in part (i) has $f(0)>0$, here $\lambda =\sqrt{2\pi }$.