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JAMIA MILLIA ISLAMIA Jamia Millia Islamia Solved Paper-2011

  • question_answer
        A function\[y=f(x)\]has a second order derivative\[f\,(x)=6(x-1).\]. If its graph passes through the point (2, 1) and at that point the tangent to the graph is\[y=3x-5,\]then the function is

    A)  \[{{(x+1)}^{3}}\]             

    B) \[{{(x-1)}^{3}}\]

    C)  \[{{(x-1)}^{2}}\]              

    D) \[{{(x+1)}^{2}}\]

    Correct Answer: B

    Solution :

                    Given, \[f\,(x)=6(x-1)\] \[\Rightarrow \]               \[f(x)=3{{(x-1)}^{2}}+c\]                        ...(i) But at (2,1),\[y=3x-5\]is tangent to\[y=f(x)\] \[\therefore \]  \[f(2)=3\] \[\therefore \]From Eq. (i), \[3=3+c\] \[\Rightarrow \]               \[c=0\] \[\therefore \]  \[f(x)=3{{(x-1)}^{2}}\] \[\Rightarrow \]               \[f(x)={{(x-1)}^{3}}+k\] Since, it passes through (2, 1). \[\therefore \]  \[1={{(2-1)}^{3}}+k\] \[\Rightarrow \]               \[k=0\]\[\therefore \]\[f(x)={{(x-1)}^{3}}\]


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