Question 6 - A-Level Maths

Exam Board	Edexcel
Module	C2 (Core Mathematics 2)
Topic	Differentiation Applications
Type	Find second derivative

$\quad \mathrm { f } ( x ) = 2 - x + 3 x ^ { \frac { 2 } { 3 } } , \quad x > 0$.
1. Find $f ^ { \prime } ( x )$ and $f ^ { \prime \prime } ( x )$.
2. Find the coordinates of the turning point of the curve $y = \mathrm { f } ( x )$.
3. Determine whether the turning point is a maximum or minimum point.
4. The points $P , Q$ and $R$ have coordinates $( - 5,2 ) , ( - 3,8 )$ and $( 9,4 )$ respectively.
5. Show that $\angle P Q R = 90 ^ { \circ }$.
Given that $P , Q$ and $R$ all lie on circle $C$,
find the coordinates of the centre of $C$,
show that the equation of $C$ can be written in the form $$x ^ { 2 } + y ^ { 2 } - 4 x - 6 y = k$$ where $k$ is an integer to be found.