A sample of a radioactive nucleus $$A$$ disintegrates to another radioactive nucleus $$B$$, which in turn disintegrates to some other stable nucleus $$C$$. Plot of a graph showing the variation of number of atoms of nucleus $$B$$ versus time is: (Assume that at $$t = 0$$, there are no $$B$$ atoms in the sample)

Rate equation for nucleus $$B$$:  $$\frac{dN_B}{dt} = \lambda_1 N_A - \lambda_2 N_B$$

Using $$N_A = N_0 e^{-\lambda_1 t}$$ with boundary condition $$N_B(0) = 0$$:

$$N_B(t) = \frac{\lambda_1 N_0}{\lambda_2 - \lambda_1} \left( e^{-\lambda_1 t} - e^{-\lambda_2 t} \right)$$

$$\text{At } t = 0 \implies N_B = 0$$

$$\text{At } t \to \infty \implies N_B \to 0$$

Since $$N_B(t) > 0$$ for all intermediate times, the curve must rise continuously from zero, reach a single global maximum, and then fall off continuously back toward zero.