Diminishing marginal returns and the optimal investment rate

If we apply the law of diminishing marginal returns it seems plausible that g(s) can be written as (4.29), which we can use to rewrite g(s) in formula (4.24).

Now we can divide V(s) by Y0 and rewrite equation (4.24) to present the value of an economy as a multiple (M(s)) of the aggregated production at present (Y0), such that M(s) = V(s)/Y0. Such a function is plotted below. 

The figure shows that the value of an economy has a maximum at the optimal investment rate. This value can be derived if we differentiate M(s) to s and solve this for dM/ds=0, like formula (4.31). 

1.4 Aggregated time accounting and quality of life