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

  • question_answer
    On the interval [0, 1], the function \[{{x}^{25}}{{(1-x)}^{75}}\] takes its maximum value at the point [IIT 1995]

    A) 0

    B) 1/2

    C) 1/3

    D) ¼

    Correct Answer: D

    Solution :

    • \[f(x)={{x}^{25}}{{(1-x)}^{75}}\]           
    • \[f'(x)={{x}^{25}}(75){{(1-x)}^{74}}(-1)+25{{x}^{24}}{{(1-x)}^{75}}\]           
    • For maxima and minima,           
    • \[-75{{x}^{25}}{{(1-x)}^{74}}+25{{x}^{24}}{{(1-x)}^{75}}=0\]           
    • Þ \[25{{x}^{24}}{{(1-x)}^{74}}[(1-x)-3x]=0\]           
    • Þ Either \[x=0\]or \[x=1\]or \[x=\frac{1}{4}\]           
    • At \[x=\frac{1}{4},\ \ f'\,\left( \frac{1}{4}-h \right)>0\]and\[f'\left( \frac{1}{4}+h \right)<0\]           
    • \[\therefore f(x)\] is maximum at \[x=\frac{1}{4}\].           
    • Trick: Check with the options.


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