<s>[INST] How many times does the function $y=e^x $ meet $y=x^2$? As you know $y=e^x$ and $y= x^2$ meet once on $x<0$. 
But I want to know whether or not they meet on $x>0$. 
Since $\lim_{x\rightarrow \infty } e^x/x^2=\infty$, if they meet once on $x>0$, they 
 must meet again. 
To summarize, my question is whether or not they meet on $x>0$. 
Thank you in advance. 
 [/INST] Note that $$e^x - x^2 = \left(e^{x/2}-x\right)\left(e^{x/2}+x\right)$$
For $x\ge 0$, the second factor is strictly positive, so consider the first factor:
$$\begin{align}e^{x/2}-x &= \left(1+\frac{x}{2}+\frac{x^2}{8} + \frac{x^3}{48} + \cdots\right) - x \\
&= 1 - \frac{x}{2}+\frac{x^2}{8} + \frac{x^3}{48}+\cdots \\[6pt]
&= \frac{1}{2} + \frac{1}{8}\left( 4-4x+x^2 \right) + \frac{x^3}{48} + \cdots \\[6pt]
&= \frac{1}{2} + \frac{1}{8} \left( 2-x \right)^2 + \frac{x^3}{48} + \cdots \\[6pt]
&\ge \frac{1}{2}
\end{align}$$
Thus, that first factor is also strictly positive, whence $e^x - x^2$ is as well. QED
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