Suppose the cost function for some product is C(x)=5√x+200C(x)=5x+200

The total cost of producing xx items is given by the function:

C(x)=5√x+200=5x12+200C(x)=5x+200=5x12+200

The average cost function is:

A(x)=C(x)x=5x12+200x=5x−12+200xA(x)=C(x)x=5x12+200x=5x−12+200x

a.)

Average cost when x=400x=400 is:

A(400)=5(400)−12+200400=5(202)−12+12=520+12=0.25+0.50=0.75$A(400)=5(400)−12+200400=5(202)−12+12=520+12=0.25+0.50=0.75$

b.)

The rate of change of average cost is:

A′(x)=ddx(5x−12+200x)=5ddx(x−12)+200ddx(x−1)=5(−12x(−12−1))+200(−1x−1−1)[ By using [1] ]=−52x−32−200x−2A′(x)=ddx(5x−12+200x)=5ddx(x−12)+200ddx(x−1)=5(−12x(−12−1))+200(−1x−1−1)[ By using [1] ]=−52x−32−200x−2

The rate of change of average cost when x=400x=400 is:

A′(400)=−52(400)−32−200(400)−2=−52(202)−32−200(202)−2=−52(20)−3−200(20)−4=−52(20)3−200(20)4=−516000−200160000=−0.0003125−0.00125=−0.0015625