```Question 633223
As a quadratic function with a positive coefficient for the term in {{{x^2}}},
it goes through a minimum.
Quadratic functions have the general form
{{{f(x)=ax^2+bx+c}}}
They graph as parabolas,
and have a minimum or maximum at {{{x=-b/2a}}}.
In the case of {{{c(x)=0.2x^2-1.3x+8.181}}} , there is a
minimum at {{{x=1.3/(2*0.2)}}} --> {{{x=1.3/0.4}}} --> {{{highlight(x=3.25)}}}
So the cost per bicycle is minimum when {{{3.25}}} hundred bicycles are built.
That is {{{highlight(325)bicycles}}}

For a quadratic function, {{{f(x)=ax^2+bx+c}}} ,
when the leading coefficient, {{{a}}}, is positive,
the function grows without bounds as {{{x^2}}} increases on both ends (positive and negative),
as the {{{ax^2}}} term overwhelms whatever value the rest of the polynomial could take.
As a consequence the function looks like a smile, with a minimum in the middle:
{{{graph(200,100,3,9,1,11,(x-6)^2+2)}}}
If {{{a<0}}} the shape of the graph is flipped and the function has a maximum:
{{{graph(200,100,3,9,-12,-2,-(x-6)^2-3)}}}
The maximum or minimum is the vertex of the parabola.
The function can be transformed algebraically from {{{f(x)=ax^2+bx+c}}} to
{{{f(x)=a((x+b/2a)^2-(b^2-4ac)/4a^2)}}}
which shows that the extreme value (maximum or minimum) occurs when
{{{x+b/2a=0}}}, when {{{x=-b/2a}}}
and the line represented by {{{x=-b/2a}}} is the axis of symmetry of the parabola that is the graph of the function.
If the expression reminds you of the quadratic formula, it is no coincidence.
The quadratic formula comes from the same
{{{f(x)=a((x+b/2a)^2-(b^2-4ac)/4a^2)}}} making {{{f(x)=0}}} .```