At time t = 0 , a material is composed of two radioactive atoms A...

The rate at which nuclei of $B$ is forming, 

$\dfrac{d{N}_{B}}{dt} = \lambda{N}_{A} - \lambda{N}_{B}$

$\dfrac{d{N}_{B}}{dt} = 2\lambda{N}_{B0}{e}^{- \lambdat} - \lambda{N}_{B}$

${e}^{- \lambdat}\left(\dfrac{d{N}_{B}}{dt} + \lambda{N}_{B}\right) = 2\lambda{N}_{B0}{e}^{- \lambdat}{e}^{\lambdat}$

$\dfrac{d}{dt}\left({N}_{B}{e}^{- \lambdat}\right) = 2\lambda{N}_{B0}$

on integrating above the equation, 
${N}_{B}{e}^{\lambdat} = 2\lambdat{N}_{B} + {N}_{{B}_{0}}$
${N}_{B}{e}^{\lambdat} = 2\lambdat{N}_{B0} + {N}_{{B}_{0}}$

${N}_{B} = {N}_{B}\left[1 + 2\lambdat\right]{e}^{- \lambdat}$
$\dfrac{d}{dt}{N}_{B} = 0$

$- \lambda\left[1 + 2\lambdat\right]{e}^{- \lambdat} + 2\lambda{e}^{- \lambdat} = 0$
${N}_{Bmax}att = \dfrac{1}{2\lambda}$