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

  • question_answer
    If at each point of the curve \[y={{x}^{3}}-a{{x}^{2}}+x+1,\]the tangent is inclined at an acute angle with the positive direction of the x-axis, then

    A) \[a>0\]

    B) \[a\le \sqrt{3}\]

    C) \[-\sqrt{3}\le a\le \sqrt{3}\]

    D) None of these

    Correct Answer: C

    Solution :

    [c] \[\frac{dy}{dx}=3{{x}^{2}}-2ax+1\] Given that \[\frac{dy}{dx}\ge 0\] or \[3{{x}^{2}}-2ax+1\ge 0\] for all \[x\] or \[D\le 0\] or \[4{{a}^{2}}-12\le 0\] or \[-\sqrt{3}\le a\le \sqrt{3}\]


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