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JEE Main & Advanced Mathematics Applications of Derivatives Question Bank Self Evaluation Test - Application of Derivatives

  • question_answer
    The profit function, in rupees, of a firm selling x items \[(x\ge 0)\] per week is given by\[P(x)=-3500+(400-x)x\]. How many items should the firm sell so that the firm has maximum profit?

    A) 400

    B) 300

    C) 200

    D) 100

    Correct Answer: C

    Solution :

    [c] \[P(x)=-3500+(400-x)x=\] \[-3500+400x-{{x}^{2}}\] On differentiating w.r.t.x, we get             \[P'(x)=400-2x\] Put \[P'(x)=0\] for maxima or minima \[\Rightarrow 400-2x=0\] \[\Rightarrow x=200\] Now \[P''(x)=-2x\] \[\Rightarrow P''(200)=-400<0\] \[\therefore \,\,\,\,\,P(x)\] is maximum at \[x=200\] Hence 200 items should the firm sell so that the firm has maximum profit.


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