Will you integrate it 2015 times? - VII

Calculus Level 3

A function $f : \mathbb{R} \rightarrow \mathbb{R}$ is differentiated 2015 times to get $g(x) = \dfrac{ d^{2015} }{dx^{2015} } \big[ f(x) \big]$ .

True or False

If $g(x) = g(-x)$ for all values of $x$ , then $f(x)$ is an odd function.

True False

1 solution

Ivan Koswara
Nov 12, 2015

Take $f(x) = 1$ everywhere. Its derivative is $0$ everywhere, and thus all of its higher derivatives are also $0$ everywhere. But $f$ is not odd.

Yes, the 0 function is both even and odd, which can complicate such considerations.