Question

The cost of manufacturing a container for frozen orange juice is $C\left(h\right)=h^2-7.4h+13.7$cents, where $h$is the height of the container in inches. Your boss claims that the containers will be the cheapest to make if they are $4$inches tall. Use Theorem 3.3 to quickly show that he is wrong.

Short answer

The claim made by the boss was wrong as it does provide the solution zero.

Step-by-step solution

Step 1. Given Information.

The function is,

$C\left(h\right)=h^2-7.4h+13.7$.

Step 2. Using the theorem.

The theorem states that, if $x=c$is the location of a local extremum of $f$, then $x=c$is a critical point of $f$.

Now, $C\left(h\right)=h^2-7.4h+13.7$

Differentiating we get,

$C\text{'}\left(h\right)=2h-7.4$

Inputting $h=4$,

$$\begin{gathered} C\text{'}\left(4\right)=2\times 4-7.4 \\ C\text{'}\left(h\right)=8-7.4 \\ C\text{'}\left(h\right)=0.6 \end{gathered}$$

The claim made by the boss was wrong as it does provide the solution zero.