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kherson [118]
3 years ago
14

A monopolist faces the demand function P = 140 − 2Q, with P denoting the market price and Q denoting the quantity demanded.

Mathematics
1 answer:
devlian [24]3 years ago
3 0

Answer:

a) The profit function is P(Q)= -13Q^3 +8Q^2-44Q

b) The derivative of the profit function is P \,'(Q)=-39Q^2 +16Q-44

c) There are no values for Q where P \, '(Q)= 0.

Step-by-step explanation:

The Profit function is defined as the difference between Revenue function and Cost function, where the Revenue function is defined as the price times the number of products.

So starting with the revenue we have:

R(Q)=PQ

where P is the market price and Q the quantity demanded, using the demand function we can write:

R(Q)=(140-2Q)Q

a) Thus the Profit function is

P(Q)=R(Q)-C(Q)\\\\P(Q)=140Q-2Q^2 -(13Q^3-10Q^2 +188Q)\\\\P(Q)=140Q-2Q^2 -13Q^3+10Q^2-188Q\\\\P(Q)= -13Q^3 +8Q^2-44Q

b) We can determine the derivative of the profit function using the result on part a).

P(Q)= -13Q^3 +8Q^2-44Q\\P \,'(Q)=-39Q^2 +16Q-44

c) We can set the profit function equal to 0

P \, '(Q)= 0

-39 Q^2 + 16 Q - 44 =0

Since we cannot factor we will have to use quadratic formula.

Q = \cfrac{-b\pm \sqrt{b^2-4ac}}{2a}

where the coefficients are

a=-39\\b = 16\\ c=-44

So we get

Q = \cfrac{-16\pm \sqrt{16^2-4(-39)(-44)}}{2(-39)}

Notice that we get a negative number inside the square root if we evaluate it.

Q = \cfrac{-16\pm \sqrt{-6608}}{2(-39)}

Since we get a negative number inside the root, then there are no real values of Q where the derivative of the profit function is equal to 0.

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Previously, cell phone keypad had a different layout than today. The numbers were related to the letters of the alphabet and the number key was pressed according to the letters that were related.

For example, if you wanted to type the letter B, you would have to press the number 2 key twice.

The relationship of numbers and letters was distributed as follows:

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The highest rate on increasing is of 12x -6y = -24; Option B is the correct answer.

The options are given in the image attached with the answer

<h3>What is a Function ?</h3>

A function is a law that relates a dependent variable and an independent variable.

It is asked among the options given , which is increasing at the highest rate.

To increase at a rate , the value of the slope , m should be > 0

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for a straight line function , the slope is given by

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Therefore the function is decreasing

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