The cost in dollars of producing x units of a product is given by: `C(x) = (4x^2-18x+3)/(3*sqrt x)` for `x >=0`   Find value of x when the marginal cost is 0.   

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The cost of producing x units as a function of x is C(x) = `(4x^2 - 18x + 3)/(3*sqrt x)`

The marginal cost of production is the derivative of C(x) with respect to x.

C'(x) = `((8x - 18)*3*sqrt x - (4x^2 - 18x + 3)*(3/(2*sqrt x)))/(9*x)`

 `((8x - 18)*3*sqrt x - (4x^2 - 18x + 3)*(3/(2*sqrt x)))/(9*x) = 0`

=> `(8x - 18)*6*x - (4x^2 - 18x + 3)*3 = 0`

=> `48x^2 - 108x - 12x^2 + 54x - 9 = 0`

=> `4x^2 - 6x - 1 = 0`

=> `x1 = (6 + sqrt(36 + 16))/8 = 3/4 + sqrt 13/4`

The other root is not considered as a negative number of units cannot be produced.

The value of x when the marginal cost is 0 is `x = 3/4 + sqrt13/4 `

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