The graph below shows a company's profit f(x), in dollars, depending on the price of erasers x, in dollars, sold by the company:

Graph of quadratic function f of x having x intercepts at ordered pairs 0, 0 and 8, 0. The vertex is at 4, 270.

Part A: What do the x-intercepts and maximum value of the graph represent? What are the intervals where the function is increasing and decreasing, and what do they represent about the sale and profit? (4 points)

Part B: What is an approximate average rate of change of the graph from x = 1 to x = 4, and what does this rate represent? (3 points)

Part C: Describe the constraints of the domain. (3 points)

A) At x-intercepts, profit is zero. At maximum, optimal price is obtained

increasing: [0,4) -> profit increase as price of erasers increase

decreasing (4,8] -> profit decrease as price of erasers increase

B) average rate of change = [f(4) - f(1)] / (4 - 1) = (270 - 120)/3 = 50

this means that, in average, a profit of 50 is made for every unitary increase in the price of erasers (at that range of prices)

C) Only those prices that makes profit positive are taken into account. And that is why the domain is [0,8]

(n^3 - n)

= n(n-1)(n+1)

since it is divisible by 3 and 2 ,because it is a multiple 3 consecutive number

if, n-1 is odd then n+1 is also odd, then it is a multiple of 9

but if, n-1 is even then n+1 is also even, then it is a multiple of 4

so, largest multiple = max(9*2, 4*3) = max(18, 12) = 18