The daily profit, 𝑃 (in $), of an oil refinery is given by 𝑃(𝑥) = −0.04𝑥2 + 10𝑥 where x isthe number of barrels of oil refined.a. How many barrels will give maximum profit and what is the maximum profit?b. Sketch the function 𝑃(𝑥)

A firm’s price in a perfectly competitive market is 1000. Its cost function is 32 C(x) = 0.01x 3x− 11+08x 96+0 , where x ≥ 0 is the number of units produced and sold. (a) Find an expression for the profit function π(x) for x ≥ 0. (b) Find all stationary points and determine the profit maximising level of output. answer to six decimal places. 1 √1−𝑥𝑥 . State your answer using (c) Using a sign diagram, determine the intervals over which π(x) is increasing and decreasing. (d) Determine the intervals over which π(x) is concave and convex. (e) Where is the point of inflection in C(x)? Give an economic interpretation of the point of inflection.

An barrels manufacturer can produce up to 300 barrels per day. The profit made fromthe sale of these barrels can be modelled by the function P (x) = − 10x2 + 3500x −66000 where P(x) is the profit in rupees and x is the number of barrels made and sold.(i) When no barrels are produce what is a profit /loss?(ii) What is the break even point ? (Zero profit point is called break even)(iii) What is the profit/loss if 175 barrels are produced(iv) What is the profit/loss if 400 barrels are produced

A cereal company estimates that its monthly cost is 𝐶(𝑥)=400𝑥2+300𝑥C(x)=400x 2 +300x and its monthly revenue is 𝑅(𝑥)=−0.6𝑥3+900𝑥2−400𝑥+700R(x)=−0.6x 3 +900x 2 −400x+700, where x is in thousands of boxes sold. The profit is the difference between the revenue and the cost.What is the profit function, P(x)?A.𝑃(𝑥)=0.6𝑥3−500𝑥2+700𝑥−700P(x)=0.6x 3 −500x 2 +700x−700B.𝑃(𝑥)=0.6𝑥3−1300𝑥2+100𝑥+700P(x)=0.6x 3 −1300x 2 +100x+700C.𝑃(𝑥)=−0.6𝑥3+1300𝑥2−100𝑥+700P(x)=−0.6x 3 +1300x 2 −100x+700D.𝑃(𝑥)=−0.6𝑥3+500𝑥2−700𝑥+700P(x)=−0.6x 3 +500x 2 −700x+700SUBMITarrow_backPREVIOUS

The organizer of a sports event estimates that if the event is announced 𝑥 days in advance, the revenueobtained will be 𝑅(𝑥) thousand dollars, where𝑅(𝑥) = 400 + 120𝑥 − 𝑥2The cost of advertising the event for 𝑥 days is 𝐶(𝑥) = 2𝑥2 + 300a. Find the profit function 𝑃(𝑥) = 𝑅(𝑥) − 𝐶(𝑥) and sketch its graph.b. How many days in advance should the event be announced in order to maximize profit? What is themaximum profit?c. What is the ratio of revenue to cost 𝑄(𝑥) = 𝑅(𝑥)𝐶(𝑥) at the optimal announcement time found in part (b)?What happens to this ratio as 𝑥 → 0? Interpret these results

onsider the profit function (Q is the quantity of goods sold)π = −12Q2 + 24Q + 53(a) Compute dπdQ and d2πdQ2 .(b) Use part (a) to find the maximum profit, and write the value of Q for whichthe maximum is attained