Cummings Carter
05/17/2024 · Primary School
1. A company has to spend \( \$ 20,000 \) in order to properly run. Additionally, this company has costs of \( \$ 6 \) per product for the first 5,000 produced units. If the company produces more than 5,000 units, the additional production cost for a unit is \( \$ 4 \). Find the production cost function \( p(x) \) for \( x \) units, and find the following values if they exist: \[ \lim _{x \rightarrow 5,000^{-}} p(x), \lim _{x \rightarrow 5,000^{+}} p(x), \lim _{x \rightarrow 5,000} p(x) \]
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Quick Answer
The production cost function \( p(x) \) is:
\[
p(x) =
\begin{cases}
20000 + 6x & \text{if } x \leq 5000 \\
4x + 30000 & \text{if } x > 5000
\end{cases}
\]
The limits are:
\[
\lim_{x \to 5000^{-}} p(x) = 50000, \quad \lim_{x \to 5000^{+}} p(x) = 50000, \quad \lim_{x \to 5000} p(x) = 50000
\]
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