Question 4 - A-Level Maths

Exam Board	AQA
Module	C1 (Core Mathematics 1)
Year	2013
Session	June
Topic	Factor & Remainder Theorem
Type	Verify factor then solve related equation

The polynomial $\mathrm { f } ( x )$ is given by $\mathrm { f } ( x ) = x ^ { 3 } - 4 x + 15$.
1. Use the Factor Theorem to show that $x + 3$ is a factor of $\mathrm { f } ( x )$.
2. Express $\mathrm { f } ( x )$ in the form $( x + 3 ) \left( x ^ { 2 } + p x + q \right)$, where $p$ and $q$ are integers.
A curve has equation $y = x ^ { 4 } - 8 x ^ { 2 } + 60 x + 7$.
1. Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
2. Show that the $x$-coordinates of any stationary points of the curve satisfy the equation $$x ^ { 3 } - 4 x + 15 = 0$$
3. Use the results above to show that the only stationary point of the curve occurs when $x = - 3$.
4. Find the value of $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ when $x = - 3$.
5. Hence determine, with a reason, whether the curve has a maximum point or a minimum point when $x = - 3$.