Question: from Bikram, Canada
A movie is attended by 200 people when the admission is $3 per ticket. For every $0.25 increase in the admission price, ten fewer will people attend. What price will maximize the revenue?
Answer: Let R = (3 + 0.25x)(200 - 10x) from which we get R = 600 + 20x - 2.5x2. Completing the square gives R = 640 - 2.5(x - 4)2. This is maximized when x = 4. The price to maximize revenue is 3 + 0.25(4) = 4.
Solution to bonus question
Since the left side is non-negative, so is the right side. Thus, 2x - 3 > 0 or x > 1.5. Squaring both sides and collecting like terms, we get 3x2 - 12x + 25 > 0 which is true for any value of x. Finally, for the left side to be non-negative, x < -4 or x > 4. Combined with the above restrictions, the smallest value of x to satisfy the inequality is 4.