Question

The marginal cost function of x units of a product is given by 2MC= 3x² -10x +3x² The cost of producing one unit is Rs. 7. Find the cost function and average cost function. 

Answer 1

Let the total cost function be (C_x)
${\displaystyle \therefore }$ Marginal cost Mc = ${\displaystyle \frac{dc}{dx}}$
${\displaystyle \therefore \frac{dc}{dx}=3x^2-10x+3}$          .........(1)

${\displaystyle \therefore c\left(x\right)=\int (3x^2-10x+3)dx}$
          ${\displaystyle =3\left(\frac{x^3}{3}\right)-10\left(\frac{x^2}{2}\right)+3x+A}$
                     C(x) = x³ - 5x² + 3x + A          ........(2)
                     If n = 1 , C(1) = 7
${\displaystyle \therefore 1-5+3+A=7}$
${\displaystyle \therefore A=7+5-4=8}$
${\displaystyle \therefore C\left(x\right)=x^3-5x^2+3x+8}$            .....(3)
Also, the average cost function is given by
${\displaystyle \frac{C\left(x\right)}{x}=x^2-5x+3+\frac{8}{x}}$          .....(4)