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JEE Main & Advanced Sample Paper JEE Main Sample Paper-33

  • question_answer
    The values of a for which the points of extremism of the function \[f\left( x \right)={{x}^{3}}-3\alpha {{x}^{2}}+3\left( {{\alpha }^{2}}-1 \right)\text{ }x+1\]lie in the interval  \[\left( -2,\text{ }4 \right)\]will be equal to

    A)  \[\left( 3,4 \right)\]       

    B)     \[\left( -4,-2 \right)\]

    C)  \[\left( -1,3 \right)\]

    D)                     \[\left( -2,-1 \right)\]

    Correct Answer: C

    Solution :

    We have \[f(x)={{x}^{3}}-3\alpha {{x}^{2}}+3({{a}^{2}}-1)x+1\]             So, \[f'(x)=3({{x}^{2}}-2\alpha x+{{\alpha }^{2}}-1)\]             \[=3(x-\alpha +1)(x-\alpha -1)\] Cleary,\[\alpha -1>-2\] and \[\alpha +1<4\] must be satisfied simultaneously, so\[\alpha \in (-1,3).\]


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