Question

the cost of manufacturing $x$ toasters in one day is given by $c(x)=-0.08x^{2}+15x - 300, 0 < x < 400$. (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. $x =$ next item

Explanation:

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

$C(x)=- 0.08x^{2}+15x - 300$, so $C'(x)=-0.16x + 15$.

Step2: Find critical values

Set $C'(x) = 0$. Then $-0.16x+15 = 0$. Solving for $x$ gives $x=\frac{15}{0.16}=93.75$.

Step3: Determine increasing and decreasing intervals

Test intervals using the critical - value. The domain is $0\lt x\lt400$. Choose a test - point in the interval $(0,93.75)$, say $x = 1$. Then $C'(1)=-0.16\times1 + 15=14.84>0$, so $C(x)$ is increasing on the interval $(0,93.75)$. Choose a test - point in the interval $(93.75,400)$, say $x = 100$. Then $C'(100)=-0.16\times100 + 15=-1<0$, so $C(x)$ is decreasing on the interval $(93.75,400)$.

Step4: Find relative extrema

Since $C(x)$ changes from increasing to decreasing at $x = 93.75$, there is a relative maximum at $x = 93.75$ and no relative minimum.

Answer:

(a) $x = 93.75$
(b) $(0,93.75)$
(c) $(93.75,400)$
(d) $x = 93.75$
(e) NONE