## 4 Special Cases  
In this section, we introduce some concrete methods based on the proposed class Equation (9). _Method 1_ Let us consider the weight functions defined by  
\[G(\tau)=\frac{\gamma}{6}\tau^{3}+2\tau^{2}+1\text{ and }H(\tau,\phi)=\frac{\gamma}{2} \tau^{3}+\left(\frac{\gamma}{6}+3\right)\tau^{2}+4\tau\phi+2\tau+\phi+1,\]  
where \(\tau=\frac{f(y_{n})}{f(x_{n})},\ \ \phi=\frac{f(z_{n})}{f(y_{n})}\) and \(\gamma\) is any free disposable parameter. It can be easily seen that the above mentioned weight functions \(G(\tau)\) and \(H(\tau,\phi)\) satisfy all the conditions of Theorem (1). Therefore, we get a new derivative-free optimal family of eighth-order methods given by  
\[\left\{ \begin{array}{l}y_{n}=x_{n}-\frac{f(x_{n})}{f[x_{n},w_{n}]},\ \ \ w_{n}=x_{n}+\beta f(x_{n})^{3},\\ z_{n}=x_{n}-\left(\frac{f(x_{n})+f(y_{n})}{f[x_{n},w_{n}]}\right)\left[\frac{ \gamma}{6}\left(\frac{f(y_{n})}{f(x_{n})}\right)^{3}+2\left(\frac{f(y_{n})}{f (x_{n})}\right)^{2}+1\right],\\ x_{n+1}=z_{n}-\frac{f(z_{n})}{f[x_{n},w_{n}]}\left[\frac{\gamma}{2}\left(\frac {f(y_{n})}{f(x_{n})}\right)^{3}+\left(\frac{\gamma}{6}+3\right)\left(\frac{f( y_{n})}{f(x_{n})}\right)^{2}+4\frac{f(y_{n})}{f(x_{n})}\frac{f(z_{n})}{f(y_{n}) }+2\frac{f(y_{n})}{f(x_{n})}+\frac{f(z_{n})}{f(y_{n})}+1\right].\end{array} \right. \tag{23}\]  
_Method 2_ Now, we consider the following weight functions  
\[G(\tau)=\frac{\tau(1-12(\mu+2)\tau)-12}{\tau(1-12\mu\tau)-12}\text{ and }H(\tau,\phi)=\frac{-24+\left(\frac{299}{3}+48\mu\right)\tau^{3}}{4(-6+6\phi+(1 2-5\tau)\tau)},\]  
where \(\tau=\frac{f(y_{n})}{f(x_{n})},\ \ \phi=\frac{f(z_{n})}{f(y_{n})}\) and \(\mu\) is any free disposable parameter. These weight functions satisfy all the conditions of Theorem (1). Therefore, we obtain another new derivative-free optimal family of eighth-order methods given by  
\[\left\{ \begin{array}{l}y_{n}=x_{n}-\frac{f(x_{n})}{f[x_{n},w_{n}]},\ \ \ w_{n}=x_{n}+\beta f(x_{n})^{3},\\ z_{n}=x_{n}-\left(\frac{f(x_{n})+f(y_{n})}{f[x_{n},w_{n}]}\right)\left[\frac{ \frac{f(y_{n})}{f(x_{n})}(1-12(\mu+2)\frac{f(y_{n})}{f(x_{n})})-12}{ \frac{f(y_{n})}{f(x_{n})}(1-12\mu\frac{f(y_{n})}{f(x_{n})})-12}\right],\\ x_{n+1}=z_{n}-\frac{f(z_{n})}{f[x_{n},w_{n}]}\left[\frac{-24+\left(\frac{299}{ 3}+48\mu\right)\frac{f(y_{n})}{f(x_{n})}}{4(-6+6\frac{f(z_{n})}{f(y_{n})}+(12 -5\frac{f(y_{n})}{f(x_{n})})\frac{f(y_{n})}{f(x_{n})})}\right].\end{array} \right. \tag{24}\]  
_Method 3_ Consider the weight functions defined by  
\[G(\tau)=\frac{6\eta-\tau+12\eta\tau^{2}+(\eta-2)\tau^{3}}{6\eta-\tau}\text{ and }H(\tau,\phi)=\frac{\tau^{2}-6\eta\left(12+25\tau^{2}\right)}{\tau^{2}+6\eta(-12 +12\phi+(24-35\tau)\tau)},\]where \(\tau=\dfrac{f(y_{n})}{f(x_{n})},\ \ \phi=\dfrac{f(z_{n})}{f(y_{n})}\) and \(\eta\) is any free disposable parameter. These weight functions also satisfy all the conditions of Theorem (1). Therefore, we get another new optimal family of eighth-order methods given by  
\[\left\{ \begin{array}{l}y_{n}=x_{n}-\dfrac{f(x_{n})}{f[x_{n},w_{n}]},\quad w_{ n}=x_{n}+\beta f(x_{n})^{3},\\ z_{n}=x_{n}-\left(\dfrac{f(x_{n})+f(y_{n})}{f[x_{n},w_{n}]}\right) \left[\dfrac{6\eta-\dfrac{f(y_{n})}{f(x_{n})}+12\eta\dfrac{f(y_{n})}{f(x_{n}) }^{2}+(\eta-2)\dfrac{f(y_{n})}{f(x_{n})}^{3}}{6\eta-\dfrac{f(y_{n})}{f(x_{n})} }\right],\\ x_{n+1}=z_{n}-\dfrac{f(z_{n})}{f[x_{n},w_{n}]}\left[\dfrac{\dfrac{f(y_{n})}{f( x_{n})}^{2}-6\eta\left(12+25\dfrac{f(y_{n})}{f(x_{n})}^{2}\right)}{\dfrac{f(y_{n}) }{f(x_{n})}^{2}+6\eta(-12+12\dfrac{f(z_{n})}{f(y_{n})}+(24-35\dfrac{f(y_{n}) }{f(x_{n})}\dfrac{f(y_{n})}{f(x_{n})})}\right].\end{array} \right. \tag{25}\]  
_Method 4_ The method by [PERSON] and [PERSON], see [2], denoted by \(KTM_{8}\), is  
\[\left\{