Functions whose derivatives grow fast

Let $c\in\R$ . Is there a twice differentiable function $f\colon \R\to\R$ such that for all $x\in\R$ :
$f'(x)>f(x)+c\,\,\,\text{and}\,\,\, f''(x)>f'(x)+c\quad ?$

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Anduril@anduril
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NawzadHogan@nawzadhogan

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Solution by Anduril

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If $c<0$ , the exponential function is a solution.
If $c=0$ (also true if $c<0$ ), $x\longmapsto exp(2x)$ works.
If $c>0$ , it is impossivle : let $g$ be $f'-f$ .

We have for every real $x$ : $g(x)>c$ and $g'(x)>c$ .

But by integrating the second inequality for $x<0$ we have : $g(x)<cx+g(0)$ so that $g(x)<0$ when $x$ approaches negative infinity.

That is contradictory with the first condition.

Note that if $f''$ is not regular enough, we cannot integrate $g'$ but we can bypass this issue by using the Mean Value Theorem.

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