Question

The cost to produce a product is modeled by the function f(x) = 5x^2 − 70x + 258, where x is the number of products produced. Complete the square to determine the minimum cost of producing this product.

5(x − 7)^2 + 13; The minimum cost to produce the product is $13.
5(x − 7)^2 + 13; The minimum cost to produce the product is $7.
5(x − 7)^2 + 258; The minimum cost to produce the product is $7.
5(x − 7)^2 + 258; The minimum cost to produce the product is $258.

Answer 1

The minimum of a quadratic function, with a positive coefficient a, is its vertex.

Let's find the x₀ coordinate.

`f(x) = 5x^2 -70x + 258\\\\x_0=(-b)/(2a)=(-(-70))/(2*5) =(70)/(10) =7`

Now we need to find y₀ coordinate. That will be the minimum of function.

`y_0=5*7^2-70*7+258=13`

So, the minimum cost to produce the product is $13

Decompose 5x^2 − 70x + 258 into multipliers

`5x^2 - 70x + 258=(5x^2-70x+245)+13=5(x^2-14+49)+13=\\=5(x-7)^2+13`

Answer: 5(x − 7)^2 + 13; The minimum cost to produce the product is $13.