Find range using calculus

10 The function f is defined by \(\mathrm { f } ( x ) = x ^ { 2 } + \frac { k } { x } + 2\) for \(x > 0\).

1. Given that the curve with equation \(y = \mathrm { f } ( x )\) has a stationary point when \(x = 2\), find \(k\).
2. Determine the nature of the stationary point.
3. Given that this is the only stationary point of the curve, find the range of f .

3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of a curve with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \frac { 2 x + 3 } { \sqrt { 4 x - 1 } } \quad x > \frac { 1 } { 4 }$$

1. Find, in simplest form, \(\mathrm { f } ^ { \prime } ( x )\).
2. Hence find the range of f.

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7. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = e ^ { - x ^ { 2 } } \left( 2 x ^ { 2 } - 3 \right) ^ { 2 }$$

1. Find the range of f
2. Show that $$\mathrm { f } ^ { \prime } ( x ) = 2 x \left( 2 x ^ { 2 } - 3 \right) \mathrm { e } ^ { - x ^ { 2 } } \left( A - B x ^ { 2 } \right)$$ where \(A\) and \(B\) are constants to be found. Given that the line \(y = k\), where \(k\) is a constant, \(k > 0\), intersects the curve at exactly two distinct points,
3. find the exact range of values of \(k\)

14. Given that $$y = \frac { \left( x ^ { 2 } - 4 \right) ^ { \frac { 1 } { 2 } } } { x ^ { 3 } } \quad x > 2$$

1. show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { A x ^ { 2 } + 12 } { x ^ { 4 } \left( x ^ { 2 } - 4 \right) ^ { \frac { 1 } { 2 } } } \quad x > 2$$ where \(A\) is a constant to be found. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a377da06-a968-438c-bec2-ae55283dae47-48_593_1134_865_395} \captionsetup{labelformat=empty} \caption{Figure 4}
   \end{figure} Figure 4 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \frac { 24 \left( x ^ { 2 } - 4 \right) ^ { \frac { 1 } { 2 } } } { x ^ { 3 } } \quad x > 2$$
2. Use your answer to part (a) to find the range of f.
3. State a reason why f-1 does not exist.

8. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = \frac { 6 x + 2 } { 3 x ^ { 2 } + 5 } , \quad x \in \mathbb { R }$$

1. Find \(\mathrm { f } ^ { \prime } ( x )\), writing your answer as a single fraction in its simplest form. The curve has two turning points, a maximum at point \(A\) and a minimum at point \(B\), as shown in Figure 2.
2. Using part (a), find the coordinates of point \(A\) and the coordinates of point \(B\).
3. State the coordinates of the maximum turning point of the function with equation $$y = \mathrm { f } ( 2 x ) + 4 \quad x \in \mathbb { R }$$
4. Find the range of the function $$\operatorname { g } ( x ) = \frac { 6 x + 2 } { 3 x ^ { 2 } + 5 } , \quad x \leqslant 0$$

8. $$\mathrm { h } ( x ) = \frac { 2 } { x + 2 } + \frac { 4 } { x ^ { 2 } + 5 } - \frac { 18 } { \left( x ^ { 2 } + 5 \right) ( x + 2 ) } , \quad x \geqslant 0$$

1. Show that \(\mathrm { h } ( x ) = \frac { 2 x } { x ^ { 2 } + 5 }\)
2. Hence, or otherwise, find \(\mathrm { h } ^ { \prime } ( x )\) in its simplest form. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{e375f6ad-4a76-42a0-b7bf-ae47e5cbdaeb-26_679_1168_733_390} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} Figure 2 shows a graph of the curve with equation \(y = \mathrm { h } ( x )\).
3. Calculate the range of \(\mathrm { h } ( x )\).

7. $$\mathrm { h } ( x ) = \frac { 2 } { x + 2 } + \frac { 4 } { x ^ { 2 } + 5 } - \frac { 18 } { \left( x ^ { 2 } + 5 \right) ( x + 2 ) } , \quad x \geqslant 0$$

1. Show that \(\mathrm { h } ( x ) = \frac { 2 x } { x ^ { 2 } + 5 }\)
2. Hence, or otherwise, find \(\mathrm { h } ^ { \prime } ( x )\) in its simplest form. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c78b0245-5c5a-407f-ad8a-602949a76e05-10_729_1235_644_351} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} Figure 2 shows a graph of the curve with equation \(y = \mathrm { h } ( x )\).
3. Calculate the range of \(\mathrm { h } ( x )\).

7. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve with equation $$\mathrm { g } ( x ) = x ^ { 2 } ( 1 - x ) \mathrm { e } ^ { - 2 x } , \quad x \geqslant 0$$

1. Show that \(\mathrm { g } ^ { \prime } ( x ) = \mathrm { f } ( x ) \mathrm { e } ^ { - 2 x }\), where \(\mathrm { f } ( x )\) is a cubic function to be found.
2. Hence find the range of g .
3. State a reason why the function \(\mathrm { g } ^ { - 1 } ( x )\) does not exist.