Deriving the golden-rule savings rate in a Solow Model
The "golden rule" is the level at which steady-state consumption is at a maximum, given the parameters of the model. Steady state consumption is

$$c^* = (1-s^*)\cdot f[k^*(s^*)] = f[k^*(s^*)]  - s^*f[k^*(s^*)] \tag{1}$$

where $0&lt;s^*&lt;1$

We also have that, at the steady state (for constant capital)

$$s^*f[k^*(s^*)] = \delta k^*(s^*) \tag{2}$$

Inserting $(2)$ in $(1)$,

$$c^* =  f[k^*(s^*)]  - \delta k^*(s^*) \tag{3}$$

We want to maximize steady-state consumption, so we take the first derivative and set it equal to zero,

$$\frac {\partial c^*}{\partial s^*} = f'[k^*(s^*)]\cdot \frac {{\rm d}k^*}{{\rm d}s^*} - \delta \frac {{\rm d}k^*}{{\rm d}s^*} =0 $$

$$\implies \big(f'[k^*(s^*)]\ - \delta \big)\frac {{\rm d}k^*}{{\rm d}s^*} =0 \implies f'[k^*(s^*)] = \delta \tag{4} $$

From the production function we have

$$f'(k) = \frac 1{\sqrt {k}} = \delta \implies k^* = \frac 1 {\delta^2}$$

The rest are evident.