Find second derivative

2. $$y = 2 x ^ { 2 } - \frac { 4 } { \sqrt { } x } + 1 , \quad x > 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving each term in its simplest form.
2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\), giving each term in its simplest form.
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4. $$y = 5 x ^ { 3 } - 6 x ^ { \frac { 4 } { 3 } } + 2 x - 3$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) giving each term in its simplest form.
2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)

10. For the curve \(C\) with equation \(y = \mathrm { f } ( x )\), \(\frac { d y } { d x } = x ^ { 3 } + 2 x - 7 .\)

1. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
   (2)
2. Show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } \geq 2\) for all values of \(x\).
   (1)
   Given that the point \(P ( 2,4 )\) lies on \(C\),
3. find \(y\) in terms of \(x\),
   (5)
4. find an equation for the normal to \(C\) at \(P\) in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers.
   (5)
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   1. continued

1. $$f ( x ) = x ^ { 3 } + 3 x ^ { 2 } + 5$$ Find

1. \(\mathrm { f } ^ { \prime \prime } ( x )\),
2. \(\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x\).

3 Given that \(y = 3 x ^ { 5 } - \sqrt { x } + 15\), find

1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).

1. Given that

$$y = \sqrt { x } - \frac { 4 } { \sqrt { x } }$$

1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
2. find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\),
3. show that $$4 x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 x \frac { \mathrm {~d} y } { \mathrm {~d} x } - y = 0 .$$

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 \(x = 3\).

8 Given that \(y = 6 x ^ { 3 } + \sqrt { x } + 3\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).

2 Given that \(y = 6 x ^ { \frac { 3 } { 2 } }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
Show, without using a calculator, that when \(x = 36\) the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) is \(\frac { 3 } { 4 }\).

10 Given tha \(y = 6 x ^ { \frac { 3 } { 2 } }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
Show, without using a calculator, that when \(x = 36\) the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) is \(\frac { 3 } { 4 }\).

12 Given tha \(y = 6 x ^ { 3 } + \sqrt { x } + 3\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).

6 Given that \(y = 6 x ^ { 3 } + \frac { 4 } { \sqrt { x } } + 5 x\), find

1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
   \(7 \quad A\) is the point \(( 5,7 )\) and \(B\) is the point \(( - 1 , - 5 )\).
3. Find the coordinates of the mid-point of the line segment \(A B\).
4. Find an equation of the line through \(A\) that is perpendicular to the line segment \(A B\), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.

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.

7 In this question you must show detailed reasoning.
A curve has equation \(y = 4 x ^ { 3 } - 6 x ^ { 2 } - 9 x + 4\).

1. Sketch the gradient function for this curve, clearly indicating the points where the gradient is zero.
2. Find the set of values of \(x\) for which the gradient function is decreasing. Give your answer using set notation.

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

8 The gradient, \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), at the point \(( x , y )\) on a curve is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 54 + 27 x - 6 x ^ { 2 }$$

1. 1. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
   2. The curve passes through the point \(P \left( - 1 \frac { 1 } { 2 } , 4 \right)\). Verify that the curve has a minimum point at \(P\).
2. 1. Show that at the points on the curve where \(y\) is decreasing $$2 x ^ { 2 } - 9 x - 18 > 0$$
   2. Solve the inequality \(2 x ^ { 2 } - 9 x - 18 > 0\).

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.

7. The curve \(y = a x ^ { 4 } + b x ^ { 3 } + 18 x ^ { 2 }\) has a point of inflection at \(( 1,11 )\).

1. Show that \(2 a + b + 6 = 0\).
2. Find the values of the constants \(a\) and \(b\) and show that the curve has another point of inflection at \(( 3,27 )\).
3. Sketch the curve, identifying all the stationary points including their nature.

6 In this question you must show detailed reasoning.

1. Given that \(y = \mathrm { e } ^ { 2 x } - 8 \mathrm { e } ^ { x } - 16 x ^ { 2 } + 7 x - 3\), determine the range of values of \(x\) for which $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } < 0$$
2. State the geometrical interpretation of your answer to part (a), in terms of the shape of the graph of \(y = \mathrm { f } ( x )\) (not the gradient of the graph).

9. The gradient, \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), at the point \(( x , y )\) on a curve is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 54 + 27 x - 6 x ^ { 2 }$$

1. 1. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
      [0pt] [2 marks]
   2. The curve passes through the point \(P \left( - 1 \frac { 1 } { 2 } , 4 \right)\). Verify that the curve has a minimum point at \(P\).
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2. 1. Show that at the points on the curve where \(y\) is decreasing $$2 x ^ { 2 } - 9 x - 18 > 0$$
   2. Solve the inequality \(2 x ^ { 2 } - 9 x - 18 > 0\).

3. 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.
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