Calc - Extreme Values - Word Problem

The profit, *P*, in dollars for selling **x **hamburgers is modelled by $\displaystyle P(x)= 2.44x - \frac {x^2}{20 000} - 5 000$, where $\displaystyle 0 \leq x \leq 35 000$. For what quantities of hamburgers is the profit increasing and decreasing?

$\displaystyle P'(x) = 2.44 - 40000x $

$\displaystyle 0 = 2.44 - 40 000x$

$\displaystyle 6.1^{-0.5} = x$

$\displaystyle P(6.1^{-0.5}) = -4999$

$\displaystyle P(0) = -5000$

$\displaystyle P(35 000) = 19 150$

Therefore, decreasing at $\displaystyle (0, -5000) $ and increasing at $\displaystyle (35 000, 19 150)$

**Textbook Answer: **

increasing - $\displaystyle 0 \leq x \leq 24 400$

decreasing - $\displaystyle 24 400 \leq x \leq 35 000$