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?

Category: QuestionsIf 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?
Editor">Editor Staff asked 11 months ago

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
 
The question was asked in an online quiz.
 
Query is from Calculus Application topic in division Application of Calculus of Mathematics – Class 12
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1 Answers
Editor">Editor Staff answered 11 months ago

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.