Will you integrate it 2015 times? - II

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 $g(x) = 0$ for all values of $x$ , then $f(x) = 0$ has finitely many solutions.

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

True False

1 solution

Pranshu Gaba
Nov 11, 2015

Let $f ^{ (n) } (x)$ denote the $n$ th derivative of $f(x)$ with respect to $x$ .

We are given that $g(x) = f^{(2015)} (x) = 0$ for all $x$ . After integrating $2015$ times, we get

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

Here, $c_{i}$ 's can be any real numbers. Note that if all the $c_{i}$ 's are zero, then $f(x) = 0$ for all $x$ , and it will have infinitely many solutions. We have found a counterexample to the statement in the problem, therefore the statement is $\boxed {\text{False} }$ . $_\square$

Will you integrate it 2015 times? - II

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

1 solution

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