Answer
$-1$
Work Step by Step
Need to find the absolute maximum value of the function
$f(x) =x-e^{x} $
$f'(x) =1-e^{x}=0$
This implies $e^{x} -1=0$
Thus $x=0$
$f'(x) $ will be increasing for all $x<0$ and $f'(x) $ will be decreasing when $x>0$.
Therefore, by first derivative test for absolute extreme values, $f(x) =x-e^{x} $
has an absolute maximum value at x = 0 and absolute maximum value of $f(x) =x-e^{x} $
is $f(0) = 0-1 = -1$ at $x=0$.
