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

  • question_answer
    If the function \[f(x)=2{{x}^{3}}-9a{{x}^{2}}\] \[+12{{a}^{2}}x+1,\]where \[a>0\] attains its maximum and minimum at p and q respectively such that \[{{p}^{2}}=q\], then a  equals [AIEEE 2003]

    A) 3

    B) 1

    C) 2

    D) \[\frac{1}{2}\]

    Correct Answer: C

    Solution :

    • \[f(x)=2{{x}^{3}}-9a{{x}^{2}}+12{{a}^{2}}x+1\]           
    • \[{f}'(x)=6{{x}^{2}}-18ax+12{{a}^{2}}\]           
    • \[{f}''(x)=12x-18a\]           
    • For maximum and minimum ,           
    • \[6{{x}^{2}}-18ax+12{{a}^{2}}=0\Rightarrow {{x}^{2}}-3ax+2{{a}^{2}}=0\]           
    • \[x=a\] or \[x=2a\], at \[x=a\] maximum and at \[x=2a\] minimum           
    • \[\because \] \[{{p}^{2}}=q\],            
    • \[\therefore \] \[{{a}^{2}}=2a\Rightarrow a=2\] or \[a=0\]           
    • But \[a>0\], therefore \[a=2.\]


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