Question

1) The profit equation for a taco stand is given by P(x) = -0.4x2 + 100x - 100, where x is the number of tacos sold, and P(x) is the profit in dollars. Use the first-derivative test to find when relative extrema occur. 2) The cost equation for a bakery is given by C(x) 10(x - 2)2 - 120(x - 2) 450, where x is the number of doughnuts made (in dozens), and C(x) is the cost in dollars. Use the first-derivative test to find when relative extrema occur

Answer 1

Answer

1)

Profit function is given by : P(x) = -0.4x² + 100x - 100

First order condition :

d(P)/dx = 0 => -0.4*2x + 100 = 0

=> x = 125 ------------------------Critical value

Lets use First derivative to test to determine whether it is a relative maximum or relative minimum.

(i) If x is little bit lesser than 125 and d(P)/dx < 0 and If x is little bit greater than 125 and d(P)/dx > 0 then it is a relative minimum.

(ii)

If x is little bit lesser than 125 and d(P)/dx > 0 and If x is little bit greater than 125 and d(P)/dx < 0 then it is a relative maximum.

Now If x < 125 like x = 124.99 then dP/dx = -0.4*2*124.99 + 100 = 0.008 > 0 and If x > 125 like x = 125.1 then dP/dx = -0.4*2*125.1 + 100 = -0.08 < 0

Hence using above test we get that x = 125 is a relative maximum.

2)

Profit function is given by : C(x) = 10(x - 2)² - 120(x - 2) + 450

First order condition :

d(C)/dx = 0 => 10*2(x - 2) - 120 = 0

=> x - 2 = 6

=> x = 8 ------------------------ Critical value

Lets use First derivative to test to determine whether it is a relative maximum or relative minimum.

(i) If x is little bit lesser than 8 and d(C)/dx < 0 and If x is little bit greater than 8 and d(C)/dx > 0 then it is a relative minimum.

(ii) If x is little bit lesser than 8 and d(C)/dx > 0 and If x is little bit greater than 8 and d(P)/dx < 0 then it is a relative maximum.

Now If x < 8 like x = 7.99 then dC/dx = 10*2(7.99 - 2) - 120 = -0.2 < 0 and If x > 8 like x = 8.1 then dC/dx = 10*2(7.99 - 2) - 120 = 2 > 0.

Hence, using above test we get that x = 8 is a relative minimum.