<s>[INST] The proof of $e^x \leq x + e^{x^2}$ How can we prove the inequality $e^x \le x + e^{x^2}$ for $x\in\mathbb{R}$? 
 [/INST] $$e^{x} = 1 + x + \frac{x^2}{2!} + \dots \le 1 + x + 2 \left(\left( \frac{x}{2} \right )^2+\left( \frac{x}{2} \right )^3 +\dots \right)\le 1 + x+\left( \frac{x}{2} \right )^2 \left( \frac {4}{2 - x}\right)$$
$$x + e^{x^2} \ge  x +1  + x^2 + \frac{x^4}{2!} = 1+ x+  \left( \frac{x}{2} \right )^2 2(2+x^2) \ge 1+ x+  4  \left( \frac{x}{2} \right )^2 $$
The inequality is true for all $\Bbb R \setminus (1,3)$, let $b \in (1,2)$, 
Also,
$$e^{(1+b)^2}+1+b \ge  e^{(1+b)^2} \ge e^{1+b}$$
hence the inequality holds for all $x\in \Bbb R$
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