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Question
The total cost function y for x units is given by y = `4x((x+2)/(x+1)) + 6`. Prove that marginal cost [MC] decreases as x increases.
Solution
Given y = `4x((x+2)/(x+1)) + 6`
y = `(4x^2 + 8x)/(x + 1) + 6`
Differentiating with respect to 'x' we get,
Marginal cost = `"dy"/"dx" = ((x+1)(8x+8)-(4x^2 + 8x)(1))/(x + 1)^2`
`= (8x^2 + 8x + 8x + 8 - 4x^2 - 8x)/(x + 1)^2`
`= (4x^2 + 8x + 8)/(x + 1)^2`
`= (4(x^2 + 2x + 2))/(x + 1)^2`
`= (4(x^2 + 2x + 1 + 1))/(x + 1)^2`
`= 4 ((x + 1)^2/(x + 1)^2 + 1/(x + 1)^2)`
Marginal Cost = `4(1 + 1/(1 + x)^2)`
Since `1/(1 + x)^2` is in the denominator, as x increases marginal cost decreases.
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