A family of pdfs or pmfs $\{g(t\theta):\theta\in\Theta\}$ for a univariate random variable $T$ with realvalued parameter $\theta$ has a monotone likelihood ratio (MLR) if, for every $\theta_2>\theta_1$, $g(t\theta_2)/g(t\theta_1)$ is a monotone (nonincreasing or nondecreasing) function of $t$ on $\{t:g(t\theta_1)>0\;\text{or}\;g(t\theta_2)>0\}$. Note that $c/0$ is defined as $\infty$ if $0< c$.
Consider testing $H_0:\theta\leq \theta_0$ versus $H_1:\theta>\theta_0$. Suppose that $T$ is a sufficient statistic for $\theta$ and the family of pdfs or pmfs $\{g(t\theta):\theta\in\Theta\}$ of $T$ has an MLR. Then for any $t_0$, the test that rejects $H_0$ if and only if $T >t_0$ is a UMP level $\alpha$ test, where $\alpha=P_{\theta_0}(T >t_0)$.
Example 1
To better understand the theorem, consider a single observation, $X$, from $\mathrm{n}(\theta,1)$, and test the following hypotheses:
$$
H_0:\theta\leq \theta_0\quad\mathrm{versus}\quad H_1:\theta>\theta_0.
$$
Then $\theta_1>\theta_0$, and the likelihood ratio test statistics would be
$$
\lambda(x)=\frac{f(x\theta_1)}{f(x\theta_0)}.
$$
And we say that the null hypothesis is rejected if $\lambda(x)>k$. To see if the distribution of the sample has MLR property, we simplify the above equation as follows:
$$
\begin{aligned}
\lambda(x)&=\frac{\frac{1}{\sqrt{2\pi}}\exp\left[\frac{(x\theta_1)^2}{2}\right]}{\frac{1}{\sqrt{2\pi}}\exp\left[\frac{(x\theta_0)^2}{2}\right]}\\
&=\exp
\left[\frac{x^22x\theta_1+\theta_1^2}{2}+\frac{x^22x\theta_0+\theta_0^2}{2}\right]\\
&=\exp\left[\frac{2x\theta_1\theta_1^22x\theta_0+\theta_0^2}{2}\right]\\
&=\exp\left[\frac{2x(\theta_1\theta_0)(\theta_1^2\theta_0^2)}{2}\right]\\
&=\exp\left[x(\theta_1\theta_0)\right]\times\exp\left[\frac{\theta_1^2\theta_0^2}{2}\right]
\end{aligned}
$$
which is increasing as a function of $x$, since $\theta_1>\theta_0$.

Figure 1. Normal Densities with $\mu=1,2$. 
By illustration, consider Figure 1. The plot of the likelihood ratio of these models is monotone increasing as seen in Figure 2, where rejecting $H_0$ if $\lambda(x)>k$ is equivalent to rejecting it if $T\geq t_0$.

Figure 2. Likelihood Ratio of the Normal Densities. 
And by factorization theorem the likelihood ratio test statistic can be written as a function of the sufficient statistics since the term, $h(x)$ will be cancelled out. That is,
$$
\lambda(t)=\frac{g(t\theta_1)}{g(t\theta_0)}.
$$
And by KarlinRubin theorem, the rejection region $R=\{t:t>t_0\}$ is a uniformly most powerful level$\alpha$ test. Where $t_0$ satisfies the following:
$$
\begin{aligned}
\mathrm{P}(T>t_0\theta_0)&=\mathrm{P}(T\in R\theta_0)\\
\alpha&=1\mathrm{P}(X\leq t_0\theta_0)\\
1\alpha&=\int_{\infty}^{t_0}\frac{1}{\sqrt{2\pi}}\exp\left[\frac{(x\theta_0)^2}{2}\right]\operatorname{d}x
\end{aligned}
$$
Hence the quantile of the $1\alpha$ probability, which is $z_{\alpha}$ is equal to $t_0$, that is $z_{\alpha}=t_0$, and thus we reject $H_0$ if $T>z_{\alpha}$.
Example 2
Now consider testing the hypotheses, $H_0:\theta\geq \theta_0$ versus $H_1:\theta< \theta_0$ using the sample $X$ (single observation) from Beta($\theta$, 2), and to be more specific let $\theta_0=4$ and $\theta_1=3$. Can we apply KarlinRubin?
Of course! Visually, we have something like in Figure 3.

Figure 3. Beta Densities Under Different Parameters. 
Note that for this test, $\theta_1<\theta_0$, and so the likelihood ratio test statistics is simplified as follows:
$$
\begin{aligned}
\lambda(x)&=\frac{f(x\theta_1=3, 2)}{f(x\theta_0=4, 2)}=\frac{\displaystyle\frac{\Gamma(\theta_1+2)}{\Gamma(\theta_1)\Gamma(2)}x^{\theta_11}(1x)^{21}}{\displaystyle\frac{\Gamma(\theta_0+2)}{\Gamma(\theta_0)\Gamma(2)}x^{\theta_01}(1x)^{21}}\\
&=\frac{\displaystyle\frac{\Gamma(5)}{\Gamma(3)\Gamma(2)}x^{2}(1x)}{\displaystyle\frac{\Gamma(6)}{\Gamma(4)\Gamma(2)}x^{3}(1x)}=\frac{\displaystyle\frac{12\Gamma(3)}{\Gamma(3)\Gamma(2)}x^{2}(1x)}{\displaystyle\frac{20\Gamma(4)}{\Gamma(4)\Gamma(2)}x^{3}(1x)}\\
&=\frac{3}{5x},
\end{aligned}
$$
which is decreasing as a function of $x$, see the plot of this in Figure 4. And we say that $H_0$ is rejected if $\lambda(x) > k$ if and only if $T < t_0$. Where $t_0$ satisfies the following equations:
$$
\begin{aligned}
\mathrm{P}(T < t_0\theta_0)&=\mathrm{P}(X < t_0\theta_0)\\
\alpha&=\int_{0}^{t_0}\frac{\Gamma(\theta_0+2)}{\Gamma(\theta_0)\Gamma(2)}x^{\theta_01}(1x)^{21}\operatorname{d}x\\
\alpha&=\int_{0}^{t_0}\frac{\Gamma(6)}{\Gamma(4)\Gamma(2)}x^{3}(1x)\operatorname{d}x.
\end{aligned}
$$
Hence the quantile of the $\alpha$ probability, $x_{\alpha}=t_0$. And thus we reject $H_0$ if $T < x_{\alpha}$.

Figure 4. Likelihood Ratio of the Beta Densities. 
Reference

Casella, G. and Berger, R.L. (2001). Statistical Inference. Thomson Learning, Inc.
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