Intersection of two functions from different classes of functions

Question

I am wondering that $e^x$ and $x^2$ intersect how many times? Is there any way to compute efficiently and is there any generalization of questions like that? For example, $e^x$ and $x^n$ or one is polynomial other one is exponential.

Answer 1

If you want answer theses questions you may consider the function $\phi_n(x)=e^x-x^n$. For $\phi_2(x)=e^x-x^2$, The successives derivatives are $\phi_2’(x)=e^x-2x$, and $\phi_2’’(x)=e^x-2>0\iff x\in(\log{2},\infty)$ then $\forall x, \phi_2’(x)\ge\phi’_2(\log{2})>0$. Therefore $phi_2$ is an increasing function; since $\phi_2(\mathbb{R})=\mathbb{R}$, then $\phi_2$ vanishes a single time which means that those two curves intersect each other a single time.

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It seems that for the general case it will be more simple if you consider the function $\psi(x)=\frac{e^x}{x^n}$