find the minimum average cost if the total cost function is C(x)=5x^2+7x+9 ?  

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pramodpandey | College Teacher | (Level 3) Valedictorian

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We have given

`C(x)=5x^2+7x+9`

Average cost function is

`AC(x)=(C(x))/x=(5x^2+7x+9)/x`

`=5x+7+9/x`

`AC'(x)=5-9/x^2`

For Minimum , AC'(x)=0

`5-9/x^2=0`

`x^2=9/5`

`x=sqrt(9/5)`

`AC''(x)=18/x^3`

`AC''(x)}_{x=sqrt(9/5)}>0`

Thus x=sqrt(9/5) will give minimum average cost.

Minimum AC(x)

`AC(sqrt(9/5))=5xxsqrt(9/5)+7+9/sqrt(9/5)`

`=2sqrt(45)+7`

`=20.42`  per unit

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