Functions problems and optimisation problems The cost of some product is $40. Currently its retail price is $60 and the quantity of sale is 300 per week. Now you want to make a promotion and find that for each $1 that you reduce the price you can sell 20 more per week. (a) Let x be price after promotion. Find the formula for benefit B(x) per week. (b) Optimize B(x) and find out the best promotion price x and benefit. What I did for part (a) is using the equation y = mx + c where y = 300, x = $60, c = $40 300 = 60m + 40 60m = 300 - 40 m = 260/ 60 m = 13/3 y = 13/3 x + 40 I wanted to know if I am on the right track so that I can proceed with the part (b)

Well, I'd use a second degree equation,

ax²+bx+c

And then profit P of increase in price would basically be the product between

(i) 300 + 20x,

which is how many products you're going to sell for a given decrease in price, and

(ii) 60 - x

which is the new price. (Though keep in mind that, to get profit you'ld have to subtract 40 of (ii) otherwise you'ld assume that the product is free to produce).

So: (i) * (ii)

Rest is up to you, but the easiest way is to use derivatives, whom you may or may not be familiar with...