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

  • question_answer
    If \[f(x)={{x}^{2}}+2bx+2{{c}^{2}}\]and \[g(x)=-{{x}^{2}}-2cx+{{b}^{2}}\] such that min \[f(x)>\] max \[g(x)\], then the relation between b and c is [IIT Screening 2003]

    A) No real value of b and c

    B) \[0<c<b\sqrt{2}\]

    C) \[|c|<\,|b|\sqrt{2}\]

    D) \[|c|\,>\,|b|\sqrt{2}\]

    Correct Answer: D

    Solution :

    • \[f(x)={{(x+b)}^{2}}+2{{c}^{2}}-{{b}^{2}}\] is minimum at \[x=-b\] and \[g(x)={{b}^{2}}+{{c}^{2}}-{{(x+c)}^{2}}\] is maximum at \[x=-c\]          
    • Þ \[2{{c}^{2}}-{{b}^{2}}>{{b}^{2}}+{{c}^{2}}\Rightarrow |c|\,>\sqrt{2}|b|\].


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