Question

the cost of manufacturing $x$ toasters in one day is given by
$c(x) = -0.05x^2 + 20x - 350$, $0 < x < 400$.
(a) list all the critical values of $c(x)$. if there are no critical values, enter none.
$x = $
(b) find the interval(s) where $c(x)$ is increasing.
(c) find the interval(s) where $c(x)$ is decreasing.
(d) list the $x$-value(s) of all relative maxima of $c(x)$. if there are no relative maxima, enter none.
$x = $
(e) list the $x$-value(s) of all relative minima of $c(x)$. if there are no relative minima, enter none.
$x = $

Explanation:

Step1: Find first derivative of $C(x)$

$C'(x) = \frac{d}{dx}(-0.05x^2 + 20x - 350) = -0.1x + 20$

Step2: Solve for critical values

Set $C'(x)=0$:
$-0.1x + 20 = 0$
$0.1x = 20$
$x = \frac{20}{0.1} = 200$

Step3: Test intervals for increasing $C(x)$

Test $x<200$ (e.g., $x=100$):
$C'(100) = -0.1(100)+20=10>0$, so $C(x)$ increases.

Step4: Test intervals for decreasing $C(x)$

Test $x>200$ (e.g., $x=300$):
$C'(300) = -0.1(300)+20=-10<0$, so $C(x)$ decreases.

Step5: Identify relative maxima/minima

$C(x)$ is a downward-opening parabola ($x^2$ coefficient negative), so critical point is a relative maximum. No relative minima exist.

Answer:

(a) $x=200$
(b) $(0, 200)$
(c) $(200, 400)$
(d) $x=200$
(e) 'NONE'