find maximum and minimum value of f(x)=x²-3x-4/x-8

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$$\begin{gathered} \mathrm{We} \mathrm{have}: \\ \mathrm{f}\left(\mathrm{x}\right)=\mathrm{y}=\frac{{\mathrm{x}}^2-3\mathrm{x}-4}{\mathrm{x}-8} \\ \Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\left(\mathrm{x}-8\right){\displaystyle \frac{\mathrm{d}}{\mathrm{dx}}}\left({\mathrm{x}}^2-3\mathrm{x}-4\right)-\left({\mathrm{x}}^2-3\mathrm{x}-4\right).{\displaystyle \frac{\mathrm{d}}{\mathrm{dx}}}\left(\mathrm{x}-8\right)}{{\left(\mathrm{x}-8\right)}^2} \\ =\frac{\left(\mathrm{x}-8\right)\left(2\mathrm{x}-3\right)-\left({\mathrm{x}}^2-3\mathrm{x}-4\right)}{{\left(\mathrm{x}-8\right)}^2} \\ =\frac{2{\mathrm{x}}^2-3\mathrm{x}-16\mathrm{x}+24-{\mathrm{x}}^2+3\mathrm{x}+4}{{\left(\mathrm{x}-8\right)}^2} \\ =\frac{{\mathrm{x}}^2-16\mathrm{x}+28}{{\left(\mathrm{x}-8\right)}^2} \\ =\frac{{\mathrm{x}}^2-14\mathrm{x}-2\mathrm{x}+28}{{\left(\mathrm{x}-8\right)}^2} \\ =\frac{\left(\mathrm{x}-2\right)\left(\mathrm{x}-14\right)}{{\left(\mathrm{x}-8\right)}^2} \\ \mathrm{For} \mathrm{maxima} \mathrm{or} \mathrm{minima}, \frac{\mathrm{dy}}{\mathrm{dx}}=0, \mathrm{hence} \\ \frac{\left(\mathrm{x}-2\right)\left(\mathrm{x}-14\right)}{{\left(\mathrm{x}-8\right)}^2}=0 \\ \Rightarrow \mathrm{x}=2 \mathrm{and} \mathrm{x}=14. \\ \mathrm{One} \mathrm{value} \mathrm{will} \mathrm{give} \mathrm{local} \mathrm{minima} \mathrm{while} \mathrm{other} \mathrm{will} \mathrm{give} \mathrm{local} \mathrm{maxima}. \\ \mathrm{f}\left(\mathrm{x}\right)=\frac{{\mathrm{x}}^2-3\mathrm{x}-4}{\mathrm{x}-8} \\ \Rightarrow \mathrm{f}\left(2\right)=\frac{4-6-4}{2-8}=1 \\ \mathrm{f}\left(14\right)=\frac{196-42-4}{14-8}=\frac{150}{6}=25 \\ \mathrm{Now} \\ \mathrm{f}\text{'}\left(\mathrm{x}\right)>0 \mathrm{when} \mathrm{x}\in \left(-\infty ,2\right)\cup \left(14,\infty \right) \mathrm{and} \mathrm{f}\text{'}\left(\mathrm{x}\right)<0 \mathrm{when} \mathrm{x}\in \left(2,14\right) \\ \mathrm{Hence} \mathrm{f}\left(\mathrm{x}\right) \mathrm{increases} \mathrm{when} \mathrm{x}\in \left(-\infty ,2\right), \mathrm{then} \mathrm{decreases} \mathrm{when} \mathrm{x}\in \left(2,14\right) \mathrm{and} \mathrm{again} \mathrm{increases} \mathrm{when} \mathrm{x}\in \left(14,\infty \right). \\ \mathrm{Hence} \mathrm{x}=2 \mathrm{is} \mathrm{point} \mathrm{of} \mathrm{local} \mathrm{maxima} \mathrm{and} \mathrm{x}=14 \mathrm{is} \mathrm{a} \mathrm{point} \mathrm{of} \mathrm{local} \mathrm{minima}. \\ \left[\mathrm{Note}: \mathrm{Here} \mathrm{global} \mathrm{maximum} \mathrm{and} \mathrm{minimum} \mathrm{value} \mathrm{cannot} \mathrm{be} \mathrm{determined}.\right] \end{gathered}$$

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