Will you integrate it 2015 times? - XI

Calculus Level 4

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

If $g(x)$ is a periodic function with a fundamental period $p$ , then what can be said about $f(x)$ ?
Choose the most appropriate option.

This problem is part of the set - Will you integrate it 2015 times? .

f(x)

is periodic, with fundamental period greater than or equal to

p

f(x)

is not necessarily periodic.

f(x)

is periodic, with fundamental period equal to

p

f(x)

is periodic, with fundamental period less than or equal to

p

f(x)

is periodic, with fundamental period less than, greater than or equal to

p

1 solution

Pranshu Gaba
Nov 12, 2015

Consider the function $g(x) = \sin x$ for all $x$ . It is periodic with a fundamental period of $2\pi$ . After integrating it $2015$ times, we get

$f(x) = -\cos x + c_{1} x^{2014} + c_{2} x^{2013} + c_{3} x^{2012} + \cdots + c_{2014} x + c_{2015}$

Here, $c_{i}$ 's are any real numbers. If we choose all of $c_{1}, c_{2}, \ldots, c_{2014}$ to be zero, then $f(x)$ is periodic with fundamental period $2\pi$ . Otherwise , $f(x)$ will not be periodic. This means $f(x)$ is not necessarily periodic. $_\square$

Will you integrate it 2015 times? - XI

This problem is part of the set - Will you integrate it 2015 times? .

1 solution

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