A function $y=f(x)$ has a second order derivative $f^{\prime \prime}(x)=6(x-1)$. If its graph passes through…
A function $y=f(x)$ has a second order derivative $f^{\prime \prime}(x)=6(x-1)$. 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
$(x-1)^2$
$(x-1)^3$
$(x+1)^3$
$(x+1)^2$
Solution
$f^{\prime \prime}(x)=6(x-1) \Rightarrow f^{\prime}(x)=3(x-1)^2+c$
and $f^{\prime}(2)=3 \Rightarrow c=0$
$\Rightarrow f(x)=(x-1)^3+k$ and $f(2)=1 \Rightarrow k=0$
$\Rightarrow f(x)=(x-1)^3$