Question

1. find the value of the absolute maximum and where it occurs, without the use of a calculator. \\( f(x) = 3 - x - e^{(1 - x)} \\) on \\( 0, 3 \\)

Explanation:

Step1: Find first derivative

$f'(x) = \frac{d}{dx}(3 - x - e^{1-x}) = -1 + e^{1-x}$

Step2: Find critical points

Set $f'(x)=0$:
$-1 + e^{1-x} = 0$
$e^{1-x}=1$
$1-x=\ln(1)=0 \implies x=1$

Step3: Evaluate $f(x)$ at critical point and endpoints

At $x=0$:
$f(0)=3-0-e^{1-0}=3-e\approx3-2.718=0.282$
At $x=1$:
$f(1)=3-1-e^{1-1}=2-1=1$
At $x=3$:
$f(3)=3-3-e^{1-3}=0-e^{-2}=\frac{-1}{e^2}\approx-0.135$

Step4: Compare values to find absolute maximum

Compare $f(0)\approx0.282$, $f(1)=1$, $f(3)\approx-0.135$

Answer:

The absolute maximum value is $1$, and it occurs at $x=1$.