If the absolute maximum value of the function f x = x 2 - 2 x + 7 e 4 x 3 - 12 x 2 - 180 x + 31 in the…

Given,

$f\left(x\right)=\left({\mathrm{x}}^2-2\mathrm{x}+7\right){\mathrm{e}}^{\left(4{\mathrm{x}}^3-12{\mathrm{x}}^2-180\mathrm{x}+31\right)}$

Now differentiating the function to find function is increasing or decreasing,

$f^{\text{'}}\left(x\right)=e^{\left(4x^3-12x^2-180x+31\right)}\left(12\left(x^2-2x-15\right)+2\left(x-1\right)\right)$

$\Rightarrow f^{\text{'}}\left(x\right)=e^{\left(4x^3-12x^2-180x+31\right)}\left(12\left(x+3\right)\left(x-5\right)+2\left(x-1\right)\right)$

$\Rightarrow {\mathrm{f}}^{\text{'}}\left(\mathrm{x}\right)<0$ for $x\in \left[-3,0\right]$

So, $f\left(x\right)$is decreasing function on $\left[-3,0\right]$

The absolute maximum value of the function $f\left(x\right)$ is at $x=-3$

$\Rightarrow \alpha =-3$