Where he used the information that $\mu \leq \mu_0$? 
This example is from casella_statistical inference book.
Where he used the information that $\mu \leq \mu_0$? 
What if  $\mu \geq \mu_0$? 
Thanks. 
 A: I have gone through the problem like this:
The LRT statistic is
\begin{align}
\lambda(\mathbf x)=\frac{\sup\limits_{H_0}L(\mu,\sigma^2\mid \mathbf x)}{\sup\limits_{H_0\cup H_1}L(\mu,\sigma^2\mid \mathbf x)}
&=\frac{\sup\limits_{\mu\le \mu_0,\sigma^2}L(\mu,\sigma^2\mid \mathbf x)}{\sup\limits_{\mu,\sigma^2}L(\mu,\sigma^2\mid \mathbf x)}
\\&=\frac{L\left(\hat{\hat\mu},\hat{\hat\sigma}^2\mid \mathbf x\right)}{L(\hat \mu,\hat\sigma^2\mid \mathbf x)}
\end{align}
, where $(\hat{\hat\mu},\hat{\hat\sigma}^2)$ is the restricted MLE of $(\mu,\sigma^2)$ when $\mu\le \mu_0$ and $(\hat\mu,\hat\sigma^2)$ is the unrestricted MLE of the same. It can be verified that 
$$\hat{\hat\mu}=\begin{cases}\hat\mu&,\text{ if }\mu\le \mu_0 \\ \mu_0 &,\text{ if }\mu>\mu_0\end{cases}$$
And that $$\hat{\hat\sigma}^2=\begin{cases} \hat\sigma^2&,\text{ if }\mu\le \mu_0\\ \frac{1}{n}\sum (x_i-\mu_0)^2 &,\text{ if }\mu>\mu_0\end{cases}$$
So the ratio becomes
\begin{align}
\lambda(\mathbf x)&=\begin{cases}1&,\text{ if }\mu\le \mu_0 \\ \\ \frac{L\left(\mu_0,\hat{\hat\sigma}^2\mid \mathbf x\right)}{L(\hat\mu,\hat\sigma^2\mid \mathbf x)}&,\text{ if }\mu>\mu_0\end{cases}
\end{align}
The case $\mu\ge \mu_0$ can be studied in exactly the same manner. For simply $\mu\ne \mu_0$, the ratio $\lambda(\cdot)$ no longer remains a piecewise function.
