If \(y=\mathrm{a} \log x+\mathrm{b} x^2+x\) has its extreme values at \(x=-1\) and \(x=2\), then the value of \(\left(\frac{\mathrm{a}}{\mathrm{b}}+\frac{\mathrm{b}}{\mathrm{a}}\right)\) is

\(\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{\mathrm{a}}{x}+2 \mathrm{~b} x+1 \Rightarrow\left(\frac{\mathrm{~d} y}{\mathrm{~d} x}\right)_{x=-1}=\) \(-\mathrm{a}-2 \mathrm{~b}+1=0 \)
\( \Rightarrow \mathrm{a}=-2 \mathrm{~b}+1 \)
\( \text { and }\left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)_{x=2}=\frac{\mathrm{a}}{2}+4 \mathrm{~b}+1=0 \)
\( \Rightarrow \frac{-2 \mathrm{~b}+1}{2}+4 \mathrm{~b}+1=0 \)
\( \Rightarrow-\mathrm{b}+4 \mathrm{~b}+\frac{3}{2}=0 \)
\( \Rightarrow 3 \mathrm{~b}=\frac{-3}{2} \Rightarrow \mathrm{~b}=\frac{-1}{2} \text { and } \mathrm{a}=2 \)
\( \Rightarrow\left(\frac{\mathrm{a}}{\mathrm{b}}+\frac{\mathrm{b}}{\mathrm{a}}\right)=\frac{-17}{4}\)