Information from derivatives

Suppose that $f:(-\infty,\infty)\rightarrow(-\infty,\infty)$ is given by $f(x) = x + \cos(x)$. Then $f'(x) = 1 -\sin(x)$, and we see that for any x, $0 \leq f'(x) \leq 2$. Thus all tangent lines to y = f(x) have non-negative slope and we expect that f does not have any local maximum points, as it never decreases.

Suppose that $f:(0,\infty)\rightarrow(-\infty,\infty)$ is given by $f(x) = x^3/10 - \ln(x)$. Then

$$f'(x) = (3/10)x^2 - x^{-1} = \frac{3x^3 - 10}{10 x},$$

which changes from negative to positive at $x = \sqrt[3]{10/3}$, so we expect f to have a minimum value exactly at $x = \sqrt[3]{10/3}$. We cannot get such a precise result by any other means.