If you only have f'(x) is it possible to differentiate the local and global extrema of f(x)?

Oct 31, 2015

If $f ' \left(x\right)$ is defined on the whole of $\mathbb{R}$ then it is possible to reconstruct $f \left(x\right)$ up to a constant offset and hence possible to distinguish local and global extrema. If not then it is not.

Explanation:

If $f ' \left(x\right)$ is known and defined on the whole of $\mathbb{R}$, then we can define

${f}_{1} \left(x\right) = {\int}_{0}^{x} f ' \left(t\right) \mathrm{dt}$

Then ${f}_{1} \left(x\right) = f \left(x\right) + C$ for some constant $C$

Hence we can evaluate ${f}_{1} \left(x\right)$ for any $x$ which is a root of $f ' \left(x\right) = 0$ in order to distinguish local and global maxima.

On the other hand, if $f ' \left(x\right)$ is not defined for some values of $x \in \mathbb{R}$, then we cannot distinguish local and global maxima.

For example, if $f ' \left(0\right)$ is not defined, then we do not know how the values for maxima and minima of $f \left(x\right)$ for $x \in \left(- \infty , 0\right)$ compare with maxima and minima of $f \left(x\right)$ for $x \in \left(0 , \infty\right)$.