Question

Here is a linear demand function: Q = 10 -0.5P. Find its price function by inverting the demand function. Then find its total revenue function by multiplying through by Q. The linear demand function Q = 400 -250P inverts into the price function P = 1.6 -0.004Q. Multiplying this by Q gives its total revenue function TR = 1.6Q -0.004. Evaluate the following expression.
Y = 5(2X + 3)2 -2X2

Answer 1

Answer:

$P = 20 - 2Q$

Explanation:

$Q = 10 - 0.5P$

Price function can be estimated by inverting the demand function.

$Q = 10 - 0.5P \\\\0.5P = 10 - Q\\P = 10/0.5 - Q/0.5 \\P = 20 - 2Q$

This is the price function.

Total revenue function can be estimated using the given formula,

$TR = P*Q \\ = (20 - 2Q) Q \\ = 20Q - 2Q^2$

The linear demand function is given by,

$Q = 400 - 250P$

Price function is given by,

$P = 1.6 - 0.004Q$

Total revenue function is thus given by,

$TR = P*Q \\ = 1.6Q - 0.004Q^2$

$Y = 5(2X+3)^2 - 2X^2 \\Y = 5(4X^2 + 9 + 12X) - 2X^2\\Y = 20X^2 + 45 + 60X - 2X^2\\Y = 18X^2 + 45 + 60X$

The derivative of Y with respect to x is,

$dY/dX = 36X + 60$

Equating this equal to 0 we get,

$36X + 60 = 0 \\36X = -60 \\X = -10/6 \\\\X= -1.66$