Question 1011848
You can determine the minimum by either putting it into vertex form (algebraic method) or by taking the derivative (calculus method).
Vertex:
Convert to vertex form by completing the square.
{{{C(x)=2(x^2-180x)+16420}}}
{{{C(x)=2(x^2-180x+8100)+16420-2(8100)}}}
{{{C(x)=2(x-90)^2+16420-16200}}}
{{{C(x)=2(x-90)^2+220}}}
So now the function is in vertex form.
The minimum of {{{C(x)=220}}} occurs when {{{x=90}}}.
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Derivative:
Find the derivative and set it equal to zero.
{{{dC/dx=4x-360=0}}}
{{{4x=360}}}
{{{x=90}}}
Then,
{{{C(90)=2(90)^2-360(90)+16420}}}
{{{C(90)=16200-32400+16420}}}
{{{C(90)=220}}}