Will you integrate it 2015 times? - III

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

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

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

Here, $c_{i}$ 's can be any real numbers. Since $f(x)$ is a polynomial of degree $2015$ , $f(x) = 0$ can have at most $2015$ real solutions. This means $f(x) = 0$ has finitely many solutions, therefore the statement in the problem is $\boxed {\text{True} }$ . $_\square$