The Cost C of a product is a function of the quantity x of the product: C(x)=x^2-400x+50.Find the quantity for which the cost is minimum.
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To find the quantity for which the cost is minimum, we need to find the value of x that corresponds to the minimum value of the function C(x).
The cost function is given as C(x) = x^2 - 400x + 50.
To find the quantity that minimizes the cost, we can take the derivative of the cost function with respect to x and set it equal to zero. Let's do that.
First, taking the derivative of C(x) with respect to x:
C'(x) = 2x - 400.
Now, setting C'(x) equal to zero and solving for x:
2x - 400 = 0.
Adding 400 to both sides of the equation:
2x = 400.
Dividing both sides of the equation by 2:
x = 200.
So, the quantity for which the cost is minimum is x = 200.
To find the quantity for which the cost is minimum, we need to determine the vertex of the quadratic function C(x) = x^2 - 400x + 50.
The vertex of a quadratic function is given by the formula x = -b/(2a), where a and b are the coefficients of the quadratic term and linear term, respectively.
In this case, the coefficient of the quadratic term is 1 (a = 1) and the coefficient of the linear term is -400 (b = -400).
Substituting these values into the formula, we can find the x-coordinate of the vertex:
x = -(-400) / (2*1) = 400/2 = 200
So, the quantity for which the cost is minimum is 200 units.