If y = 3x((x + a)/(x + b)) + 5 where, a and b are constants and a > b, be the total cost for x unit of output of a commodity. What will be the nature of marginal cost as the output increases continuously?

(a) Does not change

(b) Increases continuously

(c) Falls continuously

(d) Changes as the interval of y changes

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Query is from Calculus Application topic in division Application of Calculus of Mathematics – Class 12

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Correct answer is (c) Falls continuously

Explanation: We have y = 3x((x + a)/(x + b)) + 5

From the definition we get,

Marginal cost = dy/dx = d/dx[3*((x^2 + xa)/(x+b))] + 0

Solving it further, we get,

= [((x^2 + xa)/(x+b))]d[3]/dx+ 3 d[((x^2 + xa)/(x+b))]/dx

= 0 + 3 [(x+b)d(x^2 + xa)/dx – (x^2 + xa)d(x+b)/dx]/(x+b)^2

= 3[1+(b(a – b)/(x+b)^2)]

By problem, a > 0, b > 0 and a > b; hence from the expression of dy/dx, it is evident that dy/dx decreases as x increases.

Hence, we conclude the marginal cost falls continuously as the output increases.