Solow growth model - analytic proof that Inada conditions imply steady state capital is increasing in the savings rate
We want to prove that 

$$\frac{n+g+\delta}{s} &gt; f'(k^*)$$

Replace the left hand side with the equivalent from the expression $sf(k^*)=(n+\delta+g)k^*$, and you get:

$$ \frac{f(k^*)}{k^*} &gt; f'(k^*) $$

Cobb-Douglas case

Without loss of generality, assume $$f(k^*) = {k^*}^{\alpha}$$

Then, the above inequality is:

$$ {k^*}^{\alpha-1} &gt; \alpha{k^*}^{\alpha-1} $$

This is:

$$ 1&gt; \alpha $$

This condition represents a production function with decreasing marginal returns to capital, necessary for the Inada conditions to hold.

General case

A concave function has the following property, for any $x$ and $y$ in the domain:

$$f(y) \leq f(x) + f'(x)(y-x) $$

Rearranging, this is:

$$ \frac{f(y)- f(x)}{y-x} \leq f'(x) \hspace{1cm}, \text{for } y&gt;x $$

This is, the derivative of $f(x)$ at point $x$ is no smaller than the slope of the segment between $x$ and $y$. For the case of a strictly concave function (which is needed for the Inada conditions to hold), the inequality is strict ($&lt;$), except in the trivial case of $x = y$. 

For further confirmation, apply the above when $y=0$ and $x=k^*$ (notice here $y&lt;x$ so the sign of the above inequality reverses). Since $f(0)=0$, you get:

$$ \frac{f(k^*)}{k^*} &gt; f'(k^*) $$

which is what we wanted to demonstrate.