Find constant using stationary point

11 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 ( 3 x - 5 ) ^ { 3 } - k x ^ { 2 }\), where \(k\) is a constant. The curve has a stationary point at \(( 2 , - 3.5 )\).

1. Find the value of \(k\).
   ................................................................................................................................................. . .
2. Find the equation of the curve.
3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
4. Determine the nature of the stationary point at \(( 2 , - 3.5 )\).

11 The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { 2 } - 30 x + 6 a\), where \(a\) is a positive constant. The curve has a stationary point at \(( a , - 15 )\).

1. Find the value of \(a\).
2. Determine the nature of this stationary point.
3. Find the equation of the curve.
4. Find the coordinates of any other stationary points on the curve.

10 The gradient of a curve at the point \(( x , y )\) is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 ( x + 3 ) ^ { \frac { 1 } { 2 } } - x\). The curve has a stationary point at \(( a , 14 )\), where \(a\) is a positive constant.

1. Find the value of \(a\).
2. Determine the nature of the stationary point.
3. Find the equation of the curve.

10 A curve has equation \(y = \mathrm { f } ( x )\) and it is given that $$\mathrm { f } ^ { \prime } ( x ) = \left( \frac { 1 } { 2 } x + k \right) ^ { - 2 } - ( 1 + k ) ^ { - 2 }$$ where \(k\) is a constant. The curve has a minimum point at \(x = 2\).

1. Find \(\mathrm { f } ^ { \prime \prime } ( x )\) in terms of \(k\) and \(x\), and hence find the set of possible values of \(k\).
   It is now given that \(k = - 3\) and the minimum point is at \(\left( 2,3 \frac { 1 } { 2 } \right)\).
2. Find \(\mathrm { f } ( x )\).
3. Find the coordinates of the other stationary point and determine its nature.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

3 A curve has equation \(y = a x ^ { \frac { 1 } { 2 } } - 2 x\), where \(x > 0\) and \(a\) is a constant. The curve has a stationary point at the point \(P\), which has \(x\)-coordinate 9 . Find the \(y\)-coordinate of \(P\).

8 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } + a x + b\). The curve has stationary points at \(( - 1,2 )\) and \(( 3 , k )\). Find the values of the constants \(a , b\) and \(k\).

8 A curve \(y = \mathrm { f } ( x )\) has a stationary point at \(P ( 3 , - 10 )\). It is given that \(\mathrm { f } ^ { \prime } ( x ) = 2 x ^ { 2 } + k x - 12\), where \(k\) is a constant.

1. Show that \(k = - 2\) and hence find the \(x\)-coordinate of the other stationary point, \(Q\).
2. Find \(\mathrm { f } ^ { \prime \prime } ( x )\) and determine the nature of each of the stationary points \(P\) and \(Q\).
3. Find \(\mathrm { f } ( x )\).

10 A curve has equation \(y = \mathrm { f } ( x )\) and it is given that \(\mathrm { f } ^ { \prime } ( x ) = a x ^ { 2 } + b x\), where \(a\) and \(b\) are positive constants.

1. Find, in terms of \(a\) and \(b\), the non-zero value of \(x\) for which the curve has a stationary point and determine, showing all necessary working, the nature of the stationary point.
2. It is now given that the curve has a stationary point at \(( - 2 , - 3 )\) and that the gradient of the curve at \(x = 1\) is 9 . Find \(\mathrm { f } ( x )\).

6 A curve has a stationary point at \(\left( 3,9 \frac { 1 } { 2 } \right)\) and has an equation for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = a x ^ { 2 } + a ^ { 2 } x\), where \(a\) is a non-zero constant.

1. Find the value of \(a\).
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2. Find the equation of the curve.
3. Determine, showing all necessary working, the nature of the stationary point.

6 The curves \(C _ { 1 }\) and \(C _ { 2 }\) have equations $$y = \ln ( 4 x - 7 ) + 18 \quad \text { and } \quad y = a \left( x ^ { 2 } + b \right) ^ { \frac { 1 } { 2 } }$$ respectively, where \(a\) and \(b\) are positive constants. The point \(P\) lies on both curves and has \(x\)-coordinate 2 . It is given that the gradient of \(C _ { 1 }\) at \(P\) is equal to the gradient of \(C _ { 2 }\) at \(P\). Find the values of \(a\) and \(b\).

8 A curve has equation \(y = k x ^ { \frac { 3 } { 2 } }\) where \(k\) is a constant.
The point \(P\) on the curve has \(x\)-coordinate 4.
The normal to the curve at \(P\) is parallel to the line \(2 x + 3 y = 0\) and meets the \(x\)-axis at the point \(Q\). The line \(P Q\) is the radius of a circle centre \(P\). Show that \(k = \frac { 1 } { 2 }\).
Find the equation of the circle.