Question 765271
The minimum is the {{{C(x)}}} value that is the lowest point on a graph which is a parabola that opens up, so minimum occurs at the vertex ({{{h}}},{{{k}}}) where 


{{{h=-b/2a}}}

given:

{{{C(x) = 5x^2 - 50x + 225}}}

so, {{{a=5}}} and {{{b=-50}}}

meaning that {{{h=-(-50)/(2(5))=50/10=5}}}  

In other words, if {{{5}}} thousand autos are produced, then the cost will be at a {{{minimum}}}. 

Simply plug this value into the function to get:

{{{C(5) = 5*5^2 - 50*5 + 225}}} 

{{{C(5) = 125 - 250 + 225}}} 

 {{{C(5)=100}}}

So the minimum cost is {{{100}}} million dollars when {{{5}}} thousand autos are manufactured).


So the vertex is the point ({{{5}}},{{{100}}}). What this means is that if we graph {{{C(x) = 5x^2 - 50x + 225}}}, the lowest point on the graph is ({{{5}}},{{{100}}}).

{{{ graph( 600, 600, -10, 10, -10, 210, 5x^2 - 50x + 225) }}}