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A function $y = ⨍ (x)$ has a second order derivative $⨍^{11} (x)=6(x-1)$ . If its graph passes through the point (2, 1) and at that point the tangent to the graph is $y = 3 x - 5$ then the function is

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${(x - 1)}^{3}$

${(x + 1)}^{2}$

${(x - 1)}^{2}$

${(x + 1)}^{3}$

answer is D.

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Detailed Solution

$\begin{array}{rcl} f^{11} (x) & = & 6 (x - 1) \\ f^{1} (x) & = & \frac{6 {(x - 1)}^{2}}{2} + c \end{array}$

given $f^{1} (x) = 3 a - d$ point (2, 1)

$\begin{array}{rcl} 3 & = & 3 (1) + c \Rightarrow c = 0 \\ f^{2} (x) & = & 3 {(x - 1)}^{2} f (x) = {(x - 1)}^{3} + k x = 2 f (x) = 1 \\ 1 & = & 1 + k \Rightarrow k = 0 \Rightarrow f (x) = {(x - 1)}^{3} \end{array}$

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