Why my Maximum Likelihood Estimator of Pareto distribution compute fail, what's wrong whit it?

Question

I've got the density function of Pareto distribution in Wiki: https://en.wikipedia.org/wiki/Pareto_distribution#Parameter_estimation $$ p(x)=\left\{{\begin{matrix}0,&{\mbox{if }}xx_{{\min }}.\end{matrix}}\right. $$ Let me do some simplification, and we got the new dentisity function and two parameters $\alpha$ and $\beta$ $$ f(x)=\left\{{\begin{matrix}0,&{\mbox{if }}xx_{{\min }}.\end{matrix}}\right. $$ According to the maximum likelihood estimator theory, we have the liklihood function $$ {\mathcal {L}}(\alpha,\beta \,;\,x_{1},\ldots ,x_{n})=f(x_{1},x_{2},\ldots ,x_{n}\mid \alpha,\beta )=\prod _{i=1}^{n}{f(x_{i}\mid \alpha,\beta )}=\frac{\beta^n}{(\prod _{i=1}^{n}{x_i)}^{a}} $$ and let the derivative of $\alpha$ and $\beta$ equals to $0$

while $x

while $x>x_{min}$ we got $$ \frac {\partial \mathcal {L}}{\partial \alpha }=\frac{-\beta^n\sum_{i=1}^{n}{lnx_i} }{(\prod _{i=1}^{n}{x_i})^{a}}=0 $$ $$ \frac {\partial \mathcal {L}}{\partial \beta }=\frac{n\beta^{n-1}}{(\prod _{i=1}^{n}{x_i})^{a}}=0 $$ Solve the equation, we got $\beta =0$

Impossible! What's wrong with it??

Let me change some direction, let $$ ln{\mathcal {L}}= ln(\frac{\beta^n}{(\prod _{i=1}^{n}{x_i)}^{a}} )=nln\beta+ \sum_{i=1}^{n}{ln\frac{1}{x_i^a}} =nln\beta -\alpha \sum_{i=1}^{n}{lnx_i} $$ and let the derivative = $0$, we got $$ \frac {\partial ln\mathcal {L}}{\partial \alpha }=-\sum_{i=1}^{n}{lnx_i}=0 $$ $$ \frac {\partial ln\mathcal {L}}{\partial \beta }=\frac{n}{\beta}=0 $$ Solve the equation, we got $\beta =\infty$? Also impossible!

What's wrong with my compute, please help me, thank you very much!