# Hints for two proofs

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.

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

1. Solve the differential equation $$y''=y$$ and choose those such that $$y'\ne y$$.
2. Play with $$\sqrt{2}$$.
$$1.$$ Think about $$\sin \alpha$$;
$$2.$$ Think about $$x^2=\sqrt{3}$$.