The cost function of a product is given by `C(x)=(x^(3))/(3)-45x^(2)-900x+36` where x is the number of units produced. How many units should be produced to minimise the marginal cost ?

Answer

To find the number of units that should be produced to minimize the marginal cost, we will follow these steps:

### Step 1: Define the Cost Function
The cost function is given as:
\[ C(x) = \frac{x^3}{3} - 45x^2 - 900x + 36 \]

### Step 2: Find the Marginal Cost
The marginal cost (MC) is the derivative of the cost function with respect to \(x\):
\[ MC(x) = \frac{dC}{dx} \]
Calculating the derivative:
\[ MC(x) = \frac{d}{dx}\left(\frac{x^3}{3} - 45x^2 - 900x + 36\right) \]
Using the power rule:
\[ MC(x) = x^2 - 90x - 900 \]

### Step 3: Minimize the Marginal Cost
To find the minimum marginal cost, we need to differentiate the marginal cost function again and set it to zero:
\[ \frac{d(MC)}{dx} = 2x - 90 \]
Setting the first derivative equal to zero:
\[ 2x - 90 = 0 \]
Solving for \(x\):
\[ 2x = 90 \implies x = 45 \]

### Step 4: Verify Minimum Condition
To confirm that this point is indeed a minimum, we need to check the second derivative:
\[ \frac{d^2(MC)}{dx^2} = 2 \]
Since \(2 > 0\), this indicates that the marginal cost function is concave up at \(x = 45\), confirming that it is a minimum.

### Conclusion
The number of units that should be produced to minimize the marginal cost is:
\[ \boxed{45} \]

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