A collection of independent random variables $X_1,\ldots,X_n$ are modelled with a common distribution denoted by $$P(X_i \le x)=\begin{cases} 0, \text{if }x<0 \\ (x/\beta)^\alpha, \text{if }0\le x \le \beta \\ 1, \text{if }x>\beta \end{cases} $$ for fixed positive parameters $\alpha, \beta$. Find the PDF of $X_i$ and MLEs of $\alpha,\beta.$
What I have done
I took first order derivative of the piecewise function and get $$ f(x)=\frac{\alpha}{\beta}\left(\frac{x}{\beta}\right)^{\alpha-1},\text{ for }0\le x\le \beta. $$ Then I calculate the log-likelihood function and get $$ l(\alpha,\beta)=n \log(\alpha)-n \log(\beta)-(n\alpha-n)\log(\beta)+(\alpha-1)\sum_{i}\log(X_i). $$ But then I solve for $\alpha$ and get $\alpha = 0,$ I don't know where is going wrong.