Mathematics - Application of Derivative Question with Solution | TestHub

$f\left(x\right)=\left(3x^2+ax-2-a\right)e^x$
$f\text{'}\left(x\right)=\left(3x^2+ax-2-a\right)e^x+e^x(6x+a)=e^x\left(3x^2+(a+6)x-2\right)$

$\because x=1$ is a critical point $\therefore f^{\text{'}}\left(1\right)=0$

$\therefore 3+a+6-2=0$
$a=-7$

$\therefore f^{\text{'}}\left(x\right)=e^x\left(3x^2-x-2\right)=e^x\left(3x^2-3x+2x-2\right)=e^x(3x+2)(x-1)$

$\therefore$ maxima at $\mathrm{x}=\frac{-2 }{3}\therefore$ minima at $\mathrm{x}=1$