Does there exist a real function with domain $\Bbb{R}$ such that $f'(x)>0$ and $f''(x)+(f'(x))^2<0$ for all $x$?
Solution 1:
Let $g=1/f'$ so $g'>1,g>0\,\forall x\in \Bbb R$ which is impossible so such an $f$ does not exist.
Does there exist a real function with domain $\Bbb{R}$ such that $f'(x)>0$ and $f''(x)+(f'(x))^2<0$ for all $x$?
Solution 1:
Let $g=1/f'$ so $g'>1,g>0\,\forall x\in \Bbb R$ which is impossible so such an $f$ does not exist.