Suppose the total cost of producing x units of a certain commodity is c (x) = 6x4 - 30x3 - 54x2 + x + 2. determine the largest and smallest values of the marginal cost for 0 ≤ x ≤ 5.

To determine the marginal cost of the production, one needs to derive the given equation for the cost,

c (x) = 6x4 - 30x3 - 54x2 + x + 2

The derivative of the equation in terms of x is as reflected below.

c' (x) = 24x3 - 90x2 - 108x + 1

Equate the derivative to zero in order to determine the value of x.

c' (x) = 0 = 24x3 - 90x2 - 108 + 1

The value of x from the equation is 4.025.

Substituting x to the original equation,

c (x) = - 1254.35 and at c (0) = 2