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JEE Main & Advanced Mathematics Applications of Derivatives Question Bank Critical Thinking

  • question_answer
    N characters of information are held on magnetic tape, in batches of x characters each; the batch processing time is \[\alpha +\beta {{x}^{2}}\] seconds; \[\alpha \]and \[\beta \] are constants. The optimal value of x for fast processing is [MNR 1986]

    A) \[\frac{\alpha }{\beta }\]

    B) \[\frac{\beta }{\alpha }\]

    C) \[\sqrt{\frac{\alpha }{\beta }}\]

    D) \[\sqrt{\frac{\beta }{\alpha }}\]

    Correct Answer: C

    Solution :

    • Here number of batches \[=\frac{N}{x}\] and time per batch \[=(\alpha +\beta {{x}^{2}})\,\] second           
    • \[\therefore \]Total processing time \[T=\left( \frac{N}{x} \right)\,(\alpha +\beta {{x}^{2}})=N\left( \frac{\alpha }{x}+\beta x \right)\,\,\text{second}\]           
    • For fast processing T must be least,           
    • \[\therefore \frac{dT}{dx}=N\left( -\frac{\alpha }{{{x}^{2}}}+\beta  \right)\,;\ \ \frac{{{d}^{2}}T}{d{{x}^{2}}}=\frac{2N\alpha }{{{x}^{3}}}\]           
    • For maxima or minima of \[T,\ \ \frac{dT}{dx}=0\Rightarrow x=\sqrt{\left( \frac{\alpha }{\beta } \right)}\]           
    • For \[x=\sqrt{\left( \frac{\alpha }{\beta } \right)},\frac{{{d}^{2}}T}{d{{x}^{2}}}\]is +\[ve\ \ i.e.,>0\]           
    • \[\therefore \]T has minima for \[x=\sqrt{\left( \frac{\alpha }{\beta } \right)}\].


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