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

  • question_answer
    In the mean value theorem, \[f(b)-f(a)=(b-a)f'(c)\]if \[a=4\], \[b=9\] and \[f(x)=\sqrt{x}\] then the value of c is             [J & K 2005]

    A)            8.00

    B)            5.25

    C)                 4.00

    D)                 6.25

    Correct Answer: D

    Solution :

               \[f(x)=\sqrt{x}\]                    \[\therefore \,\,f(a)=\sqrt{4}=2,\] \[f(b)=\sqrt{9}=3\] ; \[{f}'(x)=\frac{1}{2\sqrt{x}}\]                     Also, \[{f}'(c)=\frac{f(b)-f(a)}{b-a}=\frac{3-2}{9-4}=\frac{1}{5}\]                 \[\therefore \] \[\frac{1}{2\sqrt{c}}=\frac{1}{5}\] Þ \[c=\frac{25}{4}=6.25\].

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