Determine nature of stationary points

2. In this question you must show all stages of your working. \section*{Solutions relying entirely on calculator technology are not acceptable.} The curve \(C\) has equation $$y = 27 x ^ { \frac { 1 } { 2 } } - x ^ { \frac { 3 } { 2 } } - 20 \quad x > 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving each term in simplest form.
2. Hence find the coordinates of the stationary point of \(C\).
3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and hence determine the nature of the stationary point of \(C\).

7. The curve \(C\) has equation $$y = 2 x ^ { 3 } - 5 x ^ { 2 } - 4 x + 2$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Using the result from part (a), find the coordinates of the turning points of \(C\).
3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
4. Hence, or otherwise, determine the nature of the turning points of \(C\).

6 A curve has gradient given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 2 } - 6 x + 9\). Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
Show that the curve has a stationary point of inflection when \(x = 3\).

2 A curve has equation \(y = x ^ { 5 } - 5 x ^ { 4 }\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
2. Verify that the curve has a stationary point when \(x = 4\).
3. Determine the nature of this stationary point.

1. The curve \(C\) has equation

$$y = 3 x ^ { 4 } - 8 x ^ { 3 } - 3$$

1. Find (i) \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) (ii) \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)
2. Verify that \(C\) has a stationary point when \(x = 2\)
3. Determine the nature of this stationary point, giving a reason for your answer.

1. The curve \(C\) has equation

$$y = 5 x ^ { 4 } - 24 x ^ { 3 } + 42 x ^ { 2 } - 32 x + 11 \quad x \in \mathbb { R }$$

1. Find
   1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
   2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)
2. 1. Verify that \(C\) has a stationary point at \(x = 1\)
   2. Show that this stationary point is a point of inflection, giving reasons for your answer.

6 Fig. 6 shows the curve with equation \(y = x ^ { 4 } - 6 x ^ { 2 } + 4 x + 5\). \begin{figure}[h] \end{figure}

7 The volume, \(V \mathrm {~m} ^ { 3 }\), of water in a tank at time \(t\) seconds is given by $$V = \frac { 1 } { 3 } t ^ { 6 } - 2 t ^ { 4 } + 3 t ^ { 2 } , \quad \text { for } t \geqslant 0$$

1. Find:
   1. \(\frac { \mathrm { d } V } { \mathrm {~d} t }\);
      (3 marks)
   2. \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} t ^ { 2 } }\).
      (2 marks)
2. Find the rate of change of the volume of water in the tank, in \(\mathrm { m } ^ { 3 } \mathrm {~s} ^ { - 1 }\), when \(t = 2\).
3. 1. Verify that \(V\) has a stationary value when \(t = 1\).
   2. Determine whether this is a maximum or minimum value.

7 The volume, \(V \mathrm {~m} ^ { 3 }\), of water in a tank at time \(t\) seconds is given by $$V = \frac { 1 } { 3 } t ^ { 6 } - 2 t ^ { 4 } + 3 t ^ { 2 } , \quad \text { for } t \geqslant 0$$

1. Find:
   1. \(\frac { \mathrm { d } V } { \mathrm {~d} t }\);
      (3 marks)
   2. \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} t ^ { 2 } }\).
      (2 marks)
2. Find the rate of change of the volume of water in the tank, in \(\mathrm { m } ^ { 3 } \mathrm {~s} ^ { - 1 }\), when \(t = 2\).
   (2 marks)
3. 1. Verify that \(V\) has a stationary value when \(t = 1\).
      (2 marks)
   2. Determine whether this is a maximum or minimum value.
      (2 marks)

4 A curve is defined for \(x > 0\). The gradient of the curve at the point \(( x , y )\) is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { x ^ { 2 } } - \frac { x } { 4 }$$

1. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
2. The curve has a stationary point \(M\) whose \(y\)-coordinate is \(\frac { 5 } { 2 }\).
   1. Find the \(x\)-coordinate of \(M\).
   2. Use your answers to parts (a) and (b)(i) to show that \(M\) is a maximum point.
   3. Find the equation of the curve.

10 A curve has equation \(y = \frac { \mathrm { e } ^ { x } } { x ^ { 2 } }\). Show that

1. the gradient of the curve at \(x = 1\) is - e ,
2. there is a stationary point at \(x = 2\) and determine its nature.

The curve \(y = (1 - x)(x^2 + 4x + k)\) has a stationary point when \(x = -3\).

1. Find the value of the constant \(k\). [7]
2. Determine whether the stationary point is a maximum or minimum point. [2]
3. Given that \(y = 9x - 9\) is the equation of the tangent to the curve at the point \(A\), find the coordinates of \(A\). [5]

A curve has equation \(y = 3x^3 - 7x + \frac{2}{x}\).

1. Verify that the curve has a stationary point when \(x = 1\). [5]
2. Determine the nature of this stationary point. [2]
3. The tangent to the curve at this stationary point meets the \(y\)-axis at the point \(Q\). Find the coordinates of \(Q\). [2]

For the curve \(C\) with equation \(y = x^4 - 8x^2 + 3\),

1. find \(\frac{dy}{dx}\), [2]
2. find the coordinates of each of the stationary points, [5]
3. determine the nature of each stationary point. [3]

The point \(A\), on the curve \(C\), has \(x\)-coordinate \(1\).

1. Find an equation for the normal to \(C\) at \(A\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [5]

$$f(x) = \frac{(x^2 - 3)^2}{x^3}, \quad x \neq 0.$$

1. Show that \(f(x) = x - 6x^{-1} + 9x^{-3}\). [2]
2. Hence, or otherwise, differentiate \(f(x)\) with respect to \(x\). [3]
3. Verify that the graph of \(y = f(x)\) has stationary points at \(x = \pm\sqrt{3}\). [2]
4. Determine whether the stationary value at \(x = \sqrt{3}\) is a maximum or a minimum. [3]

A cubic curve has equation \(y = x^3 - 3x^2 + 1\).

1. Use calculus to find the coordinates of the turning points on this curve. Determine the nature of these turning points. [5]
2. Show that the tangent to the curve at the point where \(x = -1\) has gradient 9. Find the coordinates of the other point, P, on the curve at which the tangent has gradient 9 and find the equation of the normal to the curve at P. Show that the area of the triangle bounded by the normal at P and the \(x\)- and \(y\)-axes is 8 square units. [8]

$$f(x) = 2 - x + 3x^{\frac{1}{2}}, \quad x > 0.$$

1. Find \(f'(x)\) and \(f''(x)\). [3]
2. Find the coordinates of the turning point of the curve \(y = f(x)\). [4]
3. Determine whether the turning point is a maximum or minimum point. [2]

A curve is defined for \(x \geq 0\) by the equation $$y = 6x - 2x^{\frac{1}{2}}$$

1. Find \(\frac{dy}{dx}\). [2 marks]
2. The curve has one stationary point. Find the coordinates of the stationary point and determine whether it is a maximum or minimum point. Fully justify your answer. [3 marks]

The curve C is defined for \(x > 0\) and has equation $$y = 3 - \frac{x}{2} - \frac{1}{3\sqrt{x}}$$

1. Find the exact \(x\)-coordinate of the stationary point giving your answer in the form \(a^b\) where \(a\) and \(b\) are rational numbers. [4]
2. Find the nature of the stationary point, justifying your answer. [2]