- The curve \(C\) has equation
$$y = \frac { 12 x ^ { 3 } ( x - 7 ) + 14 x ( 13 x - 15 ) } { 21 \sqrt { x } } \quad x > 0$$
- Write the equation of \(C\) in the form
$$y = a x ^ { \frac { 7 } { 2 } } + b x ^ { \frac { 5 } { 2 } } + c x ^ { \frac { 3 } { 2 } } + d x ^ { \frac { 1 } { 2 } }$$
where \(a , b , c\) and \(d\) are fully simplified constants.
The curve \(C\) has three turning points.
Using calculus, - show that the \(x\) coordinates of the three turning points satisfy the equation
$$2 x ^ { 3 } - 10 x ^ { 2 } + 13 x - 5 = 0$$
Given that the \(x\) coordinate of one of the turning points is 1
- find, using algebra, the exact \(x\) coordinates of the other two turning points.
(Solutions based entirely on calculator technology are not acceptable.)