Mathematics - Application of Derivative Question with Solution | TestHub

$f\left(x\right)=\alpha log\left|x\right|+\beta x^2+x$

If$x < 0$

$f\left(x\right)=\alpha log\left(-x\right)+\beta x^2+x$

$\therefore f{}^{\text{'}}\left(x\right)=\frac{-\alpha }{\left(-x\right)}+2\beta x+1$

If$x > 0$

$f\left(x\right)=\alpha \log \mathit{x}+\beta {\mathit{x}}^2+x$

$\therefore f{}^{\text{'}}\left(x\right)=\frac{\alpha }{x}+2\beta x+1$

$f^{\prime }\left(-1\right)=-\alpha -2\beta +1=0$

$f^{\prime }\left(2\right)=\frac{\alpha }{2}+4\beta +1=0$

$2f^{\prime }\left(-1\right)=-2\alpha -4\beta +2=0$

and$f^{\text{'}}\left(2\right)=\frac{\alpha }{2}+4\beta +1=0$

Adding,

$-\frac{3\alpha }{2}+3=0$

$\therefore \alpha =2$

$\therefore \beta =\frac{-1}{2}$