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JEE Main & Advanced Mathematics Applications of Derivatives Question Bank Self Evaluation Test - Application of Derivatives

  • question_answer
    The function \[f:[0,3]\to [1,29],\] defined by \[f(x)=2{{x}^{3}}-15{{x}^{2}}+36x+1\], is

    A) One-one and onto

    B) Onto but not one-one

    C) One-one but not onto

    D) Neither one-one nor onto

    Correct Answer: B

    Solution :

    [b] \[f(x)=2{{x}^{3}}-15{{x}^{2}}+36x+1\] \[f'(x)=6{{x}^{2}}-30x+36=6(x-2)(x-3)\] Thus, \[f(x)\] is increasing in \[[0,2]\] and decreasing in \[[2,3]\]. Therefore \[f(x)\] is many-one. \[f(0)=1;f(2)=29;f(3)=28\] Range is \[[1,29]\]. Hence, \[f(x)\] is many-one-onto.


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