Given
\[y\text{ }=\text{ }a\text{ }log\text{ }x\text{ }+\text{ }b{{x}^{2}}~+\text{ }x\]
On differentiating we get
Given that extreme values exist at
\[x\text{ }=\text{ }1,\text{ }2\]
\[a\text{ }+\text{ }2b\text{ }=\text{ }-\text{ }1\text{ }\ldots \ldots \text{ }\left( 1 \right)\]
\[a\text{ }+\text{ }8b\text{ }=\text{ }-\text{ }2\text{ }\ldots \ldots \text{ }\left( 2 \right)\]
Solving (1) and (2) we get
\[a\text{ }=\text{ }-\text{ }2/3\text{ }b\text{ }=\text{ }-\text{ }1/6\]