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JEE Main & Advanced Sample Paper JEE Main - Mock Test - 30

  • question_answer
    If the function \[f(x)=2{{x}^{3}}-9a{{x}^{2}}+12{{a}^{2}}x+1,\] \[a>0\]and f (x) is max. and min. at p and q respectively such that \[{{p}^{2}}=q\] then a =        

    A) 2p                    

    B) q

    C) 2                     

    D) p

    Correct Answer: C

    Solution :

    [c]:\[f(x)=2{{x}^{3}}-9a{{x}^{2}}+12{{a}^{2}}x+1\] \[f'(x)=6{{x}^{2}}-18ax+12{{a}^{2}}\] For maxima or minima \[6({{x}^{2}}-3ax+2{{a}^{2}})=0\] \[\Rightarrow \]\[{{x}^{2}}-3ax+2{{a}^{2}}=0\Rightarrow x=a,2a\] \[f''(x)=12x-18a\] \[f''(a)=12a-18a=-6a<0\] \[f''(2a)=12.2a-18a=6a>0\] \[\therefore \]At \[x=a,f(x)\]is maximum and at \[x=2a,f(x)\]is minimum \[\therefore \]\[p=a,q=2a\] Given \[{{p}^{2}}=q\] \[\Rightarrow {{a}^{2}}=2a\Rightarrow a=2\]                 \[[\because a>0]\]


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