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

  • question_answer
    Let \[P(x)={{a}_{0}}+{{a}_{1}}{{x}^{2}}+{{a}_{2}}{{x}^{4}}+.....+{{a}_{n}}{{x}^{2n}}\] be a polynomial in a real variable x with\[0<{{a}_{0}}<{{a}_{1}}<{{a}_{2}}<....<{{a}_{n}}\]. The function P(x) has

    A) Neither a maximum nor a minimum

    B) Only one maximum

    C) Only one minimum

    D) Only one maximum and only one minimum

    Correct Answer: C

    Solution :

    [c] The given polynomial is \[p(x)={{a}_{0}}+{{a}_{1}}{{x}^{2}}+{{a}_{2}}{{x}^{4}}+....+{{a}_{n}}{{x}^{2n}},x\in R\] and \[0<{{a}_{0}}<{{a}_{1}}<{{a}_{2}}<.....<{{a}_{n}}.\] Here, we observe that all coefficients of different powers of x, i.e., \[{{a}_{0}},{{a}_{1}},{{a}_{2}},.....,{{a}_{n}},\] are positive. Also, only even powers of x are involved. Therefore, P(x) cannot have any maximum value. Moreover, P(x) is minimum, when x = 0, i.e., \[{{a}_{0}}.\] Therefore, P(x) has only one minimum.


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