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JEE Main & Advanced Sample Paper JEE Main - Mock Test - 31

  • question_answer
    If the function \[f(x)={{x}^{4}}+b{{x}^{2}}+8x+1\] has a horizontal tangent and a point of inflection for the same value of x, then the value of b is

    A) \[-6\]             

    B)        \[-3\]                    

    C) \[3\]                    

    D)        \[6\]

    Correct Answer: A

    Solution :

         [a]  \[f(x)={{x}^{4}}+b{{x}^{2}}+8x+1\] \[\Rightarrow \,\,\,\,\,f'(x)=4{{x}^{3}}+2bx+8\] \[\Rightarrow \,\,\,\,\,f'(x)=12{{x}^{2}}+2b\] For point of inflection, \[f''(x)=0\] \[\Rightarrow \,\,\,12{{x}^{2}}+2b=0\Rightarrow 6{{x}^{2}}+b=0\] \[\Rightarrow \,\,\,\,-6{{x}^{2}}=b\]                   ?.(1) For horizontal tangent, \[f'(x)=0\] \[\Rightarrow \,\,\,4{{x}^{3}}+2bx+8=0\] \[\Rightarrow \,\,\,4{{x}^{3}}+(-12{{x}^{2}})x+8=0\]   \[[\therefore \,\,\,\,b=-6{{x}^{2}}]\] \[\Rightarrow \,\,\,\,\,{{x}^{3}}=1\]        \[\Rightarrow \,\,\,\,\,x=1\] \[\therefore \,\,\,b=-6{{x}^{2}}=-6(1)=-6\] 


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