<s>[INST] Proof about exp function, inequality I need to proof the inequality, that $e^x > 1+x^2$ for positive $x$?
Thanks
Taylor doesn't work
Easy to proof for  (0,1]. So one can assume x>1 in his proof
 [/INST] If $x > 0$, then
$$
e^{x} - 1 - x^{2}
  = x - \frac{x^{2}}{2} + \sum_{k=3}^{\infty} \frac{x^{k}}{k!} 
  \geq x - \frac{x^{2}}{2} + \frac{x^{3}}{6}
  = x(1 - \tfrac{1}{2}x + \tfrac{1}{6}x^{2}).
$$
The quadratic $1 - x/2 + x^{2}/6$ has discriminant $(\frac{1}{2})^{2} - 4\cdot\frac{1}{6} < 0$, and since it's positive at $x = 0$, it's positive for all real $x$.
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