To determine the minimum cost of producing the product using the function [tex]\( f(x) = 5x^2 - 70x + 253 \)[/tex], we will complete the square for the quadratic expression. Here is a detailed step-by-step solution:
1. Factor out the coefficient of [tex]\( x^2 \)[/tex] from the quadratic term:
[tex]\[
f(x) = 5x^2 - 70x + 253
\][/tex]
[tex]\[
f(x) = 5(x^2 - 14x) + 253
\][/tex]
2. Complete the square inside the parentheses:
- Take the coefficient of [tex]\( x \)[/tex], which is [tex]\(-14\)[/tex], divide it by 2 to get [tex]\(-7\)[/tex], and then square it to get [tex]\( 49 \)[/tex].
- Add and subtract [tex]\( 49 \)[/tex] inside the parentheses to maintain the equality of the expression.
[tex]\[
f(x) = 5(x^2 - 14x + 49 - 49) + 253
\][/tex]
[tex]\[
f(x) = 5((x - 7)^2 - 49) + 253
\][/tex]
3. Distribute the 5 back into the expression:
[tex]\[
f(x) = 5(x - 7)^2 - 5 \cdot 49 + 253
\][/tex]
[tex]\[
f(x) = 5(x - 7)^2 - 245 + 253
\][/tex]
[tex]\[
f(x) = 5(x - 7)^2 + 8
\][/tex]
4. Identify the minimum cost:
- The quadratic term [tex]\((x - 7)^2\)[/tex] is always non-negative and its minimum value is 0, which occurs when [tex]\( x = 7 \)[/tex].
- Therefore, the minimum value of the function [tex]\( f(x) \)[/tex] is achieved when the term [tex]\((x - 7)^2\)[/tex] equals 0.
[tex]\[
\text{Minimum cost} = 5(0) + 8 = 8
\][/tex]
Thus, the minimum cost to produce the product is [tex]\( \boxed{8} \)[/tex].
