Question 394690
Avg Cost/bike = {{{C(x)=0.1x^2-0.4x+7.898}}}
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This is a parabola with its "a" coefficient 0.1 being positive so the curve opens upward indicating that the vertex is a minimum. vertex is located at x=-b/(2a)= -(-0.4)/(2*0.1)=2, so the shop should build 200 bikes to minimize average cost
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Another way to do this is to take the first derivative of the avg cost function

{{{C'(x)=0.2x-0.4}}} this gives the instantenous slope and we want to find the extremas which occur when the instantenous slope=0 (also known as critical points)
0.2x-0.4=0
x=2 or 200 bikes since x is in hundreds of bikes.
to determine if this is a minimum or maximum extrema, take the second derivative

C"(x)=0.2 since this is positive at the critical point (and all points in this case) this means that it curves upward and the critical point is a minimum