Finding the Maximum with Calculus, second order condition.

Question

Question: "At a price of $8$ dollars per icket, a musical theatre group can fill every seat in the theatre, which has a capacity of $1,500$. for every additional dollar charged, the number of people buying the ticket decreases by $75$. Use calculus to find what ticket price maximized revenue. Be sure to check your second order condition."

Normally I would fill this out guess and check till I find the correct answer. How can I use calculus to make it more efficent. I know that the function is

Revenue = $(8 + x)(1,500 - 75x)$

Answer 1

Hint:

Expand, calculate the derivative $R'(x)$ and set it equal to zero, which will give a linear equation. Solve for $x$ and then plug it into $R(x)$ to find the maximum revenue.

Answer 2

$$R=(8+x) \times (1,500-75x)$$ $$R=12000-600x+1500x-75x^2$$ $$R=-75x^2+9100x+12000$$ Now take the derivative $R'$ using the generalized power rule. Set $R'$ to $0$ and solve for $x$. This is your extrema.

Next, pick your critical numbers, and use $R$ and $R'$ to check that the extremum is a maximum. There can only be one extremum here, but in many functions there will be multiple ones. Always be sure to check.