The theater make the maximum profit at d = $25. Then the maximum profit of the theatre is $ 900.
What is differentiation?
The rate of change of a function with respect to the variable is called differentiation. It may be increasing or decreasing.
A community theater uses the function P(d) = (− 4d + 40) (d − 40) to model the profit (in dollars) expected on a weekend when the tickets to a comedy show are priced at d dollars each.
Then the maximum profit of the theatre will be
The function is P(d) = (− 4d + 40) (d − 40)
Differentiate the function with respect to d and put it equal to zero for maximum or minimum profit.
[tex]\begin{aligned} \dfrac{\mathrm{d} }{\mathrm{d} d}P(d) &= 0\\\\\dfrac{\mathrm{d} }{\mathrm{d} d}(- 4d + 40) (d - 40) &= 0\\\\(-4d+40) -4 (d-40) &= 0\\\\-8d + 200 &= 0\\\\d &= 25 \end{aligned}[/tex]
Then the checking for maximum or minimum, again differentiate, we have
[tex]\begin{aligned} \dfrac{\mathrm{d} }{\mathrm{d} d}P(d) &= \dfrac{\mathrm{d} }{\mathrm{d} d}(- 4d + 40) (d - 40) \\\\\dfrac{\mathrm{d} }{\mathrm{d} d}P(d) &= \dfrac{\mathrm{d} }{\mathrm{d} d}(-8d + 200) \\\\\dfrac{\mathrm{d} }{\mathrm{d} d}P(d) &= -8\\\\ \dfrac{\mathrm{d} }{\mathrm{d} d}P(d) & < 0\end{aligned}[/tex]
The value is less than zero hence maximum value will occur at d = 25.
Then maximum profit will be
P(d) = (− 4×25 + 40) (25 − 40)
P(d) = (− 100 + 40) (−15)
P(d) = (− 60) (− 15)
P(d) = $ 900
More about the differentiation link is given below.
https://brainly.com/question/24062595
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