document.write( "Question 394690: Aki's Bicycle Designs has determined that when x hundred bicycles are built, the average cost per bicycle is given by $\"C%28x%29=0.1x%5E2-0.4x%2B7.898\"$ , where C(x) is in hundreds of dollars. How many bicycles should the shop build to minimize the average cost per bicycle?\r
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Algebra.Com's Answer #280029 by jrfrunner(365) $\"\"$ $\"About$

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Avg Cost/bike = $\"C%28x%29=0.1x%5E2-0.4x%2B7.898\"$
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\n" ); document.write( "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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\n" ); document.write( "Another way to do this is to take the first derivative of the avg cost function\r
\n" ); document.write( "\n" ); document.write( " $\"C%27%28x%29=0.2x-0.4\"$ this gives the instantenous slope and we want to find the extremas which occur when the instantenous slope=0 (also known as critical points)
\n" ); document.write( "0.2x-0.4=0
\n" ); document.write( "x=2 or 200 bikes since x is in hundreds of bikes.
\n" ); document.write( "to determine if this is a minimum or maximum extrema, take the second derivative\r
\n" ); document.write( "\n" ); document.write( "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
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