Stationary Points of Rational Functions

4. $$\mathrm { g } ( x ) = \frac { x ^ { 4 } + x ^ { 3 } - 7 x ^ { 2 } + 8 x - 48 } { x ^ { 2 } + x - 12 } , \quad x > 3 , \quad x \in \mathbb { R }$$

1. Given that $$\frac { x ^ { 4 } + x ^ { 3 } - 7 x ^ { 2 } + 8 x - 48 } { x ^ { 2 } + x - 12 } \equiv x ^ { 2 } + A + \frac { B } { x - 3 }$$ find the values of the constants \(A\) and \(B\).
2. Hence, or otherwise, find the equation of the tangent to the curve with equation \(y = \mathrm { g } ( x )\) at the point where \(x = 4\). Give your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be determined.
   (5)
   

9 The curve \(C\) has equation $$y = \frac { x ^ { 2 } - 2 x + \lambda } { x + 1 }$$ where \(\lambda\) is a constant. Show that the equations of the asymptotes of \(C\) are independent of \(\lambda\). Find the value of \(\lambda\) for which the \(x\)-axis is a tangent to \(C\), and sketch \(C\) in this case. Sketch \(C\) in the case \(\lambda = - 4\), giving the exact coordinates of the points of intersection of \(C\) with the \(x\)-axis.

The curve \(C\) has equation $$y = \frac { x ( x + 1 ) } { ( x - 1 ) ^ { 2 } }$$

1. Obtain the equations of the asymptotes of \(C\).
2. Show that there is exactly one point of intersection of \(C\) with the asymptotes and find its coordinates.
3. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence
   1. find the coordinates of any stationary points of \(C\),
   2. state the set of values of \(x\) for which the gradient of \(C\) is negative.
   3. Draw a sketch of \(C\).

6 The curve \(C\) has equation $$y = \frac { x ^ { 2 } - 3 x - 7 } { x + 1 }$$

1. Obtain the equations of the asymptotes of \(C\).
2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } > 1\) at all points of \(C\).
3. Draw a sketch of \(C\).

The curve \(C\) has equation $$y = \frac { x ^ { 2 } + \lambda x - 6 \lambda ^ { 2 } } { x + 3 }$$ where \(\lambda\) is a constant such that \(\lambda \neq 1\) and \(\lambda \neq - \frac { 3 } { 2 }\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and deduce that if \(C\) has two stationary points then \(- \frac { 3 } { 2 } < \lambda < 1\).
2. Find the equations of the asymptotes of \(C\).
3. Draw a sketch of \(C\) for the case \(0 < \lambda < 1\).
4. Draw a sketch of \(C\) for the case \(\lambda > 3\).

9 The curve \(C\) with equation $$y = \frac { a x ^ { 2 } + b x + c } { x - 1 }$$ where \(a , b\) and \(c\) are constants, has two asymptotes. It is given that \(y = 2 x - 5\) is one of these asymptotes.

1. State the equation of the other asymptote.
2. Find the value of \(a\) and show that \(b = - 7\).
3. Given also that \(C\) has a turning point when \(x = 2\), find the value of \(c\).
4. Find the set of values of \(k\) for which the line \(y = k\) does not intersect \(C\).

10 The curve \(C\) has equation \(y = \frac { 2 x ^ { 2 } - 3 x - 2 } { x ^ { 2 } - 2 x + 1 }\). State the equations of the asymptotes of \(C\). Show that \(y \leqslant \frac { 25 } { 12 }\) at all points of \(C\). Find the coordinates of any stationary points of \(C\). Sketch \(C\), stating the coordinates of any intersections of \(C\) with the coordinate axes and the asymptotes.

6 The curve \(C\) has equation $$y = \frac { x ^ { 2 } } { k x - 1 }$$ where \(k\) is a positive constant.

1. Obtain the equations of the asymptotes of \(C\).
2. Find the coordinates of the stationary points of \(C\).
3. Sketch \(C\).

The curve \(C\) has equation $$y = \frac { x ^ { 2 } + q x + 1 } { 2 x + 3 } ,$$ where \(q\) is a positive constant.

1. Obtain the equations of the asymptotes of \(C\).
2. Find the value of \(q\) for which the \(x\)-axis is a tangent to \(C\), and sketch \(C\) in this case.
3. Sketch \(C\) for the case \(q = 3\), giving the exact coordinates of the points of intersection of \(C\) with the \(x\)-axis.
4. It is given that, for all values of the constant \(\lambda\), the line $$y = \lambda x + \frac { 3 } { 2 } \lambda + \frac { 1 } { 2 } ( q - 3 )$$ passes through the point of intersection of the asymptotes of \(C\). Use this result, with the diagrams you have drawn, to show that if \(\lambda < \frac { 1 } { 2 }\) then the equation $$\frac { x ^ { 2 } + q x + 1 } { 2 x + 3 } = \lambda x + \frac { 3 } { 2 } \lambda + \frac { 1 } { 2 } ( q - 3 )$$ has no real solution if \(q\) has the value found in part (ii), but has 2 real distinct solutions if \(q = 3\).

The curve \(C\) has equation $$y = \frac { ( x - 2 ) ( x - a ) } { ( x - 1 ) ( x - 3 ) } ,$$ where \(a\) is a constant not equal to 1,2 or 3 .

1. Write down the equations of the asymptotes of \(C\).
2. Show that \(C\) meets the asymptote parallel to the \(x\)-axis at the point where \(x = \frac { 2 a - 3 } { a - 2 }\).
3. Show that the \(x\)-coordinates of any stationary points on \(C\) satisfy $$( a - 2 ) x ^ { 2 } + ( 6 - 4 a ) x + ( 5 a - 6 ) = 0$$ and hence find the set of values of \(a\) for which \(C\) has stationary points.
4. Sketch the graph of \(C\) for
   1. \(a > 3\),
   2. \(2 < a < 3\).

8 The curve \(C\) has equation \(y = \frac { 2 x ^ { 2 } + k x } { x + 1 }\), where \(k\) is a constant.

1. Find the set of values of \(k\) for which \(C\) has no stationary points.
2. For the case \(k = 4\), find the equations of the asymptotes of \(C\) and sketch \(C\), indicating the coordinates of the points where \(C\) intersects the coordinate axes.

The curve \(C\) has equation $$y = \frac { a x ^ { 2 } + b x + c } { x + 4 }$$ where \(a\), \(b\) and \(c\) are constants. It is given that \(y = 2 x - 5\) is an asymptote of \(C\).

1. Find the values of \(a\) and \(b\).
2. Given also that \(C\) has a turning point at \(x = - 1\), find the value of \(c\).
3. Find the set of values of \(y\) for which there are no points on \(C\).
4. Draw a sketch of the curve with equation $$y = \frac { 2 ( x - 7 ) ^ { 2 } + 3 ( x - 7 ) - 2 } { x - 3 }$$ [You should state the equations of the asymptotes and the coordinates of the turning points.]

9 The function f is defined by $$f ( x ) = \frac { x ^ { 2 } + 2 x + 2 } { x ^ { 2 } }$$

1. Write down the equations of the two asymptotes to the curve \(y = \mathrm { f } ( x )\).
2. By considering the expression \(x ^ { 2 } + 2 x + 2\) :
   1. show that the graph of \(y = \mathrm { f } ( x )\) does not intersect the \(x\)-axis;
   2. find the non-real roots of the equation \(\mathrm { f } ( x ) = 0\).
3. 1. Show that, if the equation \(\mathrm { f } ( x ) = k\) has two equal roots, then $$4 - 8 ( 1 - k ) = 0$$
   2. Deduce that the graph of \(y = \mathrm { f } ( x )\) has exactly one stationary point and find its coordinates.

9 A curve \(C\) has equation $$y = \frac { 2 } { x ( x - 4 ) }$$

1. Write down the equations of the three asymptotes of \(C\).
2. The curve \(C\) has one stationary point. By considering an appropriate quadratic equation, find the coordinates of this stationary point.
   (No credit will be given for solutions based on differentiation.)
3. Sketch the curve \(C\).

8 The curve \(C\) has equation \(y = \frac { 2 x ^ { 2 } + k x } { x + 1 }\), where \(k\) is a constant. Find the set of values of \(k\) for which \(C\) has no stationary points. For the case \(k = 4\), find the equations of the asymptotes of \(C\) and sketch \(C\), indicating the coordinates of the points where \(C\) intersects the coordinate axes.