Find second derivative

1 Given that \(\mathrm { f } ( x ) = 6 x ^ { 3 } - 5 x\), find

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

1. The function f, defined for \(x \in \mathbb { R } , x > 0\), is such that

$$\mathrm { f } ^ { \prime } ( x ) = x ^ { 2 } - 2 + \frac { 1 } { x ^ { 2 } }$$

1. Find the value of \(\mathrm { f } ^ { \prime \prime } ( x )\) at \(x = 4\).
2. Given that \(\mathrm { f } ( 3 ) = 0\), find \(\mathrm { f } ( x )\).
3. Prove that f is an increasing function.

4 A model helicopter takes off from a point \(O\) at time \(t = 0\) and moves vertically so that its height, \(y \mathrm {~cm}\), above \(O\) after time \(t\) seconds is given by $$y = \frac { 1 } { 4 } t ^ { 4 } - 26 t ^ { 2 } + 96 t , \quad 0 \leqslant t \leqslant 4$$

1. Find:
   1. \(\frac { \mathrm { d } y } { \mathrm {~d} t }\);
      (3 marks)
   2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } }\).
      (2 marks)
2. Verify that \(y\) has a stationary value when \(t = 2\) and determine whether this stationary value is a maximum value or a minimum value.
   (4 marks)
3. Find the rate of change of \(y\) with respect to \(t\) when \(t = 1\).
4. Determine whether the height of the helicopter above \(O\) is increasing or decreasing at the instant when \(t = 3\).

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

1. Show that the curve has no stationary points.
2. Show that the curve has exactly one point of inflection.

10 The function f is defined by $$f ( x ) = x ^ { 2 } + 2 \cos x \text { for } - \pi \leq x \leq \pi$$ Determine whether the curve with equation \(y = \mathrm { f } ( x )\) has a point of inflection at the point where \(x = 0\) Fully justify your answer.
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9 A function f is defined for all real values of \(x\) as $$f ( x ) = x ^ { 4 } + 5 x ^ { 3 }$$ The function has exactly two stationary points when \(x = 0\) and \(x = - \frac { 15 } { 4 }\)
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1. 1. Find \(\mathrm { f } ^ { \prime \prime } ( x )\)
      9
2. (ii) Determine the nature of the stationary points.
   Fully justify your answer.
   9
3. State the range of values of \(x\) for which $$f ( x ) = x ^ { 4 } + 5 x ^ { 3 }$$ is an increasing function.
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4. A second function g is defined for all real values of \(x\) as $$\mathrm { g } ( x ) = x ^ { 4 } - 5 x ^ { 3 }$$ 9
5. 1. State the single transformation which maps f onto g .
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6. (ii) State the range of values of \(x\) for which g is an increasing function.