JEE Main & Advanced Mathematics Applications of Derivatives Question Bank Rolle's theorem Lagrange's mean value theorem

  • question_answer
    The function \[f(x)=x(x+3){{e}^{-(1/2)x}}\] satisfies all the conditions of  Rolle's theorem in [?3, 0]. The value of c is

    A)            0

    B)            ?1

    C)            ? 2

    D)            ? 3

    Correct Answer: C

    Solution :

               To determine 'c' in Rolle's theorem, \[f'(c)=0\].            Here\[f'(x)=({{x}^{2}}+3x){{e}^{-(1/2)x}}.\left( -\frac{1}{2} \right)+(2x+3){{e}^{-(1/2)x}}\]                                    \[={{e}^{-(1/2)x}}\left\{ -\frac{1}{2}({{x}^{2}}+3x)+2x+3 \right\}\]                                    \[=-\frac{1}{2}{{e}^{-(x/2)}}\{{{x}^{2}}-x-6\}\]                    \[\therefore f'(c)=0\Rightarrow {{c}^{2}}-c-6=0\Rightarrow c=3,\,-2,\]                    But \[c=3\notin [-3,\,0].\]

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