Determine nature of stationary points

10. The curve \(C\) has equation $$y = 12 x ^ { \frac { 5 } { 4 } } - \frac { 5 } { 18 } x ^ { 2 } - 1000 , \quad x > 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
2. Hence find the coordinates of the stationary point on \(C\).
3. Use \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) to determine the nature of this stationary point.

4. The curve \(C\) has equation \(y = 4 x \sqrt { x } + \frac { 48 } { \sqrt { x } } - \sqrt { 8 } , x > 0\)

1. Find, simplifying each term,
   1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
   2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)
2. Use part (a) to find the exact coordinates of the stationary point of \(C\).
3. Determine whether the stationary point of \(C\) is a maximum or minimum, giving a reason for your answer.

10. A curve \(C\) has equation $$y = 4 x ^ { 3 } - 9 x + \frac { k } { x } \quad x > 0$$ where \(k\) is a constant.
The point \(P\) with \(x\) coordinate \(\frac { 1 } { 2 }\) lies on \(C\).
Given that \(P\) is a stationary point of \(C\),

1. show that \(k = - \frac { 3 } { 2 }\)
2. Determine the nature of the stationary point at \(P\), justifying your answer. The curve \(C\) has a second stationary point.
3. Using algebra, find the \(x\) coordinate of this second stationary point.
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2. A curve has equation $$y = x ^ { 3 } - x ^ { 2 } - 16 x + 2$$

1. Using calculus, find the \(x\) coordinates of the stationary points of the curve.
2. Justify, by further calculus, the nature of all of the stationary points of the curve.

9 The equation of a curve is \(y = 24 \sqrt { x } - 8 x ^ { \frac { 3 } { 2 } } + 16\).

1. Find \(\frac { \mathrm { dy } } { \mathrm { dx } }\).
2. Find the coordinates of the turning point.
3. Determine the nature of the turning point.

13 The curve with equation \(\mathrm { y } = \mathrm { px } + \frac { 8 } { \mathrm { x } ^ { 2 } } + \mathrm { q }\), where \(p\) and \(q\) are constants, has a stationary point at \(( 2,7 )\).

1. Determine the values of \(p\) and \(q\).
2. Find \(\frac { d ^ { 2 } y } { d x ^ { 2 } }\).
3. Hence determine the nature of the stationary point at (2, 7).

3 The depth of water, \(y\) metres, in a tank after time \(t\) hours is given by $$y = \frac { 1 } { 8 } t ^ { 4 } - 2 t ^ { 2 } + 4 t , \quad 0 \leqslant t \leqslant 4$$

1. Find:
   1. \(\frac { \mathrm { d } y } { \mathrm {~d} t }\);
   2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } }\).
2. Verify that \(y\) has a stationary value when \(t = 2\) and determine whether it is a maximum value or a minimum value.
3. 1. Find the rate of change of the depth of water, in metres per hour, when \(t = 1\).
   2. Hence determine, with a reason, whether the depth of water is increasing or decreasing when \(t = 1\).

6 At the point \(( x , y )\), where \(x > 0\), the gradient of a curve is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - \frac { 4 } { x ^ { 2 } } - 11$$ The point \(P ( 2,1 )\) lies on the curve.

1. 1. Verify that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) when \(x = 2\).
      (l mark)
   2. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) when \(x = 2\).
   3. Hence state whether \(P\) is a maximum point or a minimum point, giving a reason for your answer.
2. Find the equation of the curve.

3 The curve with equation \(y = x ^ { 5 } + 20 x ^ { 2 } - 8\) passes through the point \(P\), where \(x = - 2\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Verify that the point \(P\) is a stationary point of the curve.
3. 1. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at the point \(P\).
   2. Hence, or otherwise, determine whether \(P\) is a maximum point or a minimum point.
4. Find an equation of the tangent to the curve at the point where \(x = 1\).

11 It is given that for the continuous function \(g\)

• \(g ^ { \prime } ( 1 ) = 0\)
• \(\mathrm { g } ^ { \prime } ( 4 ) = 0\)
• \(\mathrm { g } ^ { \prime \prime } ( x ) = 2 x - 5\)

11

1. Determine the nature of each of the turning points of \(g\)
   Fully justify your answer.
   11
2. Find the set of values of \(x\) for which \(g\) is an increasing function.

2 A curve has equation \(y = \mathrm { f } ( x )\) The curve has a point of inflection at \(x = 7\)
It is given that \(\mathrm { f } ^ { \prime } ( 7 ) = a\) and \(\mathrm { f } ^ { \prime \prime } ( 7 ) = b\), where \(a\) and \(b\) are real numbers. Identify which one of the statements below must be true.
Circle your answer.
\(\mathrm { f } ^ { \prime } ( 7 ) \neq 0\)
\(\mathrm { f } ^ { \prime } ( 7 ) = 0\)
\(\mathrm { f } ^ { \prime \prime } ( 7 ) \neq 0\)
\(\mathrm { f } ^ { \prime \prime } ( 7 ) = 0\)