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Please do not give me the answers. I only want hints to approach these two proof problems I'm struggling with:

  1. There exists some differentiable function $f(x)$ such that $f'(x)\ne f(x)$ but $f''(x)=f(x)$
  2. There exists some $x\in \mathbb R$ such that $x^2\in \mathbb \{R -\mathbb Q\}$ while $x^4\in \mathbb R$

For the first one I started out defining $f'$ as a limit but I couldn't come up with any way to utilize this for my proof. For the second one I've tried doing it by cases, considering $x<0, x=0, x>0$. But I didn't get anywhere doing this.

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    $\begingroup$ 1. Think of trigonometric functions. 2. Let $y=x^2$ and answer the analogous problem. $\endgroup$ – David G. Stork Jan 19 at 21:07
  • $\begingroup$ 1. Think exponentials and functions built from exponentials $\endgroup$ – DMcMor Jan 19 at 21:09
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Hint.

  1. Solve the differential equation $y''=y$ and choose those such that $y'\ne y$.
  2. Play with $\sqrt{2}$.
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Hint

$1.$ Think about $\sin \alpha$;

$2.$ Think about $x^2=\sqrt{3}$.

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