Sum

The cost C of producing x articles is given as C = x^{3}-16x^{2 }+ 47x. For what values of x, with the average cost is decreasing'?

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#### Solution

C = x^{3} - 16x^{2 }+ 47x.

Average cost C_{A} = `"C"/"x" = ("x"^3 - 16"x"^2 + 47"x")/"x"`

∴ C_{A } = x^{2} - 16x + 47

Differentiating w.r.t. x

`"dC"_"A"/"dx" = 2"x" - 16`

C_{A }is decreasing if `"dC"_"A"/"dx" < 0`

i.e. 2x - 16 < 0

i.e. 2x < 16

i.e. x < 8

∴ Average cost C_{A }is decreasing for x < 8.

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