Likelihood Function How do we write the likelihood function of a problem that consists of more than one x variable?  
(In my case it is x1, x2 and x3)
 A: Let me do a couple of worked out examples. Suppose one has a sample of 3 independent and identically distributed random variables $X_1, X_2$ and $X_3$. Let's say the distribution is exponential for the sake of being concrete:
$$f(x;\lambda)=\lambda e^{-\lambda x} \;\; , \text{for } \lambda >0 \;.$$
The likelihood function is then 
$$\begin{eqnarray}L(\lambda;x_1,x_2,x_3) & = & f(x_1;\lambda)\times f(x_2;\lambda)\times f(x_3;\lambda) \\
& = & \lambda e^{-\lambda x_1}\lambda e^{-\lambda x_2}\lambda e^{-\lambda x_3} \\ 
& = & \lambda^3 e^{-\lambda (x_1+x_2+x_3)} 
\end{eqnarray}$$
which is the product of three density functions for the values $x_1,x_2,x_3$ your random variables take in your sample. If from this likelihood function you want to build an estimator for the parameter $\lambda$, you seek the maximum of the likelihood function. It is customary however to work with the loglikelihood function since it is easier to handle:
$$\begin{eqnarray}\log L(\lambda;x_1,x_2,x_3) & = & \log f(x_1;\lambda) + \log f(x_2;\lambda) + \log f(x_3;\lambda) \\
& = & \log\left(\lambda^3 e^{-\lambda (x_1+x_2+x_3)}\right) \\
& = & 3\log\lambda -\lambda (x_1+x_2+x_3)\end{eqnarray}$$
Finding the maximum of this function can be done by computing the derivative with respect to $\lambda$ and putting it equal to zero. This will lead to an equation which you can solve easily in this case:
$$\begin{eqnarray}\frac{d\log L(\lambda;x_1,x_2,x_3)}{d\lambda} & = & 0\\
\frac{d}{d\lambda}\left(3\log\lambda -\lambda (x_1+x_2+x_3)\right) & = & 0 \\
\frac{3}{\lambda} -(x_1+x_2+x_3) & = & 0 \\
\frac{3}{\lambda} & = & x_1+x_2+x_3 \\
\frac{1}{\lambda} & = & \frac{x_1+x_2+x_3}{3} \\
\lambda & = & \frac{3}{x_1+x_2+x_3}
\end{eqnarray}$$
If you want to check that this critical point does indeed correspond to a maximum, you can check if the second derivative of the loglikelihood function is negative. I leave that as an exercise to you.

If by three variables, you mean three jointly distributed variables of which you sample $n$ triplets, then the following is what you are looking for:
If your density function is of the form $f(X;\theta)$ with $X=(X_1,X_2,X_3)$ and $\theta$ a vector of parameters, and if you have $n$ data points $x_i=(x_{i,1},x_{i,2},x_{i,3})$, then you just write
$$L(\theta;x_1,\ldots,x_n) = \prod_{i=1}^n f(x_i;\theta) = \prod_{i=1}^n 
f(x_{i,1},x_{i,2},x_{i,3};\theta) \; .$$
