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

  • question_answer
    The number of solutions of the equation \[3\tan x+{{x}^{3}}=2\,\,in\left( 0,\frac{\pi }{4} \right).\] is

    A) 1

    B) 2

    C) 3

    D) Infinite

    Correct Answer: A

    Solution :

    [a] Let \[f(x)=3\tan \,x+{{x}^{3}}-2\]. Then \[f'(x)=3\,\,{{\sec }^{2}}x+3{{x}^{2}}>0\]. Hence, f(x) increases. Also, \[f(0)=-2\] and \[f\left( \frac{\pi }{4} \right)>0\]. So, by intermediate value theorem, \[f(c)=2\] for some \[c\in \left( 0,\frac{\pi }{4} \right)\]. Hence, \[f(x)=0\] has only one root.


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