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

  • question_answer
    Let \[f:(0,\,+\infty )\to R\] and \[F(x)=\int_{0}^{x}{f(t)\,dt}\]. If \[F({{x}^{2}})={{x}^{2}}(1+x)\], then \[f(4)\] equals [IIT Screening 2001]

    A) \[\frac{5}{4}\]

    B) 7

    C) 4

    D) 2

    Correct Answer: C

    Solution :

    • \[{{x}^{2}}(1+x)\,=\int_{0}^{{{x}^{2}}}{\,\,\,f(t)\,dt}.\]           
    • Differentiating w.r.t. x , \[2x(1+x)+{{x}^{2}}=f({{x}^{2}})\,.\,2x\]           
    • Þ  \[f({{x}^{2}})=1+x+\frac{x}{2},\,x>0\]           
    • Putting \[x=2,\] \[f(4)=1+2+\frac{2}{2}=4.\]


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