$\require{\cancel}$ $\require{\stix[upint]}$

Qtn No.	1	Total
Marks	11	11
Score

Question 1 Code: 9709/13/M/J/18/11, Topic: Differentiation, Integration

 

The diagram shows part of the curve $y=(x+1)^{2}+(x+1)^{-1}$ and the line $x=1$. The point $A$ is the minimum point on the curve.

$\text{(i)}$ Show that the $x$-coordinate of $A$ satisfies the equation $2(x+1)^{3}=1$ and find the exact value of $\displaystyle\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}$ at $A$. $[5]$

$\text{(ii)}$ Find, showing all necessary working, the volume obtained when the shaded region is rotated through $360^{\circ}$ about the $x$-axis. $[6]$