solve the problem.\nthe cost, in millions of dollars, for a company to manufacture x thousand automobiles is…

solve the problem.\nthe cost, in millions of dollars, for a company to manufacture x thousand automobiles is given by the function $c(x)=3x^{2}-18x + 72$. find the number of automobiles that must be produced to minimize cost.\n(a) 9 thousand automobiles\n(b) 3 thousand automobiles\n(c) 45 thousand automobiles\n(d) 6 thousand automobiles

solve the problem.\nthe cost, in millions of dollars, for a company to manufacture x thousand automobiles is given by the function $c(x)=3x^{2}-18x + 72$. find the number of automobiles that must be produced to minimize cost.\n(a) 9 thousand automobiles\n(b) 3 thousand automobiles\n(c) 45 thousand automobiles\n(d) 6 thousand automobiles

Answer

Explanation:

Step1: Identify the function type

The cost function $C(x)=3x^{2}-18x + 72$ is a quadratic function in the form $y = ax^{2}+bx + c$, where $a = 3$, $b=-18$, $c = 72$.

Step2: Recall the formula for the x - coordinate of the vertex

For a quadratic function $y=ax^{2}+bx + c$, the x - coordinate of the vertex (which gives the minimum or maximum value) is $x=-\frac{b}{2a}$.

Step3: Substitute the values of a and b

Substitute $a = 3$ and $b=-18$ into the formula $x=-\frac{b}{2a}$. We get $x=-\frac{-18}{2\times3}$.

Step4: Simplify the expression

$x=\frac{18}{6}=3$.

Answer:

B. 3 thousand automobiles