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JEE Main & Advanced Mathematics Applications of Derivatives Question Bank Critical Thinking

  • question_answer
    If the function \[f(x)={{x}^{3}}-6a{{x}^{2}}+5x\]satisfies the conditions of Lagrange's mean value theorem for the interval [1, 2] and the tangent to the curve \[y=f(x)\]at \[x=\frac{7}{4}\]is parallel to the chord that joins the points of intersection of the curve with the ordinates \[x=1\] and \[x=2\]. Then the value of \[a\]is [MP PET 1998]

    A) \[\frac{35}{16}\]

    B) \[\frac{35}{48}\]

    C) \[\frac{7}{16}\]                   

    D) \[\frac{5}{16}\]

    Correct Answer: B

    Solution :

    • \[f(b)=f(2)=8-24a+10=18-24a\]                   
    • \[f(a)=f(1)=1-6a+5=6-6a\]                   
    • \[f'(x)=3{{x}^{2}}-12ax+5\]                   
    • From Lagrange's mean value theorem,                   
    • \[f'(x)=\frac{f(b)-f(a)}{b-a}\]\[=\frac{18-24a-6+6a}{2-1}\]                   
    • \[\therefore f'(x)=12-18a\]                   
    • At  \[x=\frac{7}{4},\ 3\times \frac{49}{16}-12a\times \frac{7}{4}+5=12-18a\]                   
    • Þ \[3a=\frac{147}{16}-7\]Þ \[3a=\frac{35}{16}\] Þ \[a=\frac{35}{48}\].


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