Question

The total cost function for production and marketing of a product is given by C(x) = `(3x^2)/4 − 7x + 3,` where x is the number of units produced.

Find the level of output (number of units produced) for which MC = AC.

Answer 1

Total Cost Function: C(x) = `(3x^2)/4 − 7x + 3`

Marginal Cost is the derivative of the Total Cost:

MC = `d/dx [C(x)] = d/dx((3x^2)/4 − 7x + 3)`

MC = `3/4(2x) − 7 = (3x)/2 − 7`

Average Cost is the Total Cost divided by x:

AC = `(C(x))/x = ((3x^2)/4 − 7x + 3)/x`

AC = `(3x)/4 − 7 + 3/x`

Solve for MC = AC

Equate the expressions from Step 1 and Step 2:

`(3x)/2 −7 = (3x)/4 −7 + 3/x`   ...[Add 7 to both sides]

`(3x)/2 = (3x)/4 + 3/x`

Multiply the entire equation by 4x to clear the denominators:

`4x ((3x)/2) = 4x((3x)/4) + 4x(3/x)`

6x² = 3x² + 12

Subtract 3x² from both sides:

3x² = 12

x² = 4

x = 2