How much should a retailer mark up his goods so that when he has a 50% off sale, the resulting prices will reflect prices will still reflect a 75% markup, (on his cost)

250%

Step-by-step explanation:

Let initial Markup % = x

Sales price = 100% + x = (1 + x)%

50% off sale = (1+x)% - (1+x)% /2

Final price after 50% off sale = 1 + 0.75 = 1.75

Hence,

50% off sale = 75% price markup

0.5(1 + x) = 1.75

0.5 + 0.5x = 1.75

0.5x = 1.75 - 0.5

x = 1.25 / 0.5

x = 2.5

Hence, Markup on initial price should be 1.25 = (2.5 * 100%) = 250%

f(x) has value 150 at x= 0

and 0 at x= 3

so d(f(x))/dt = slope = -150/3 = -50