If the absolute maximum value of the function f(x) = ( x ^ 2 - 2x +...

Given,

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

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

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

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

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

So, $f\left(x\right)is decreasing function on \left[- 3, 0\right]</p><p>The absolute maximum value$ of the functionf\left(x\right)$is at$x = - 3$</p><p>$⇒ \alpha = - 3