Curve from derivative information

4 A function f is defined for \(x \in \mathbb { R }\) and is such that \(\mathrm { f } ^ { \prime } ( x ) = 2 x - 6\). The range of the function is given by \(\mathrm { f } ( x ) \geqslant - 4\).

1. State the value of \(x\) for which \(\mathrm { f } ( x )\) has a stationary value.
2. Find an expression for \(\mathrm { f } ( x )\) in terms of \(x\).

9. The curve \(C\) with equation \(y = \mathrm { f } ( x )\) passes through the point \(( 5,65 )\). Given that \(\mathrm { f } ^ { \prime } ( x ) = 6 x ^ { 2 } - 10 x - 12\),

1. use integration to find \(\mathrm { f } ( x )\).
2. Hence show that \(\mathrm { f } ( x ) = x ( 2 x + 3 ) ( x - 4 )\).
3. In the space provided on page 17, sketch \(C\), showing the coordinates of the points where \(C\) crosses the \(x\)-axis. \includegraphics[max width=\textwidth, alt={}, center]{c0db3fe8-62ec-41e3-acaf-66b2c7b2754d-11_76_40_2646_1894}

1. The curve \(C\) has equation \(y = \mathrm { f } ( x )\), where

$$f ^ { \prime } ( x ) = ( x - 3 ) ( 3 x + 5 )$$ Given that the point \(P ( 1,20 )\) lies on \(C\),

1. find \(\mathrm { f } ( x )\), simplifying each term.
2. Show that $$f ( x ) = ( x - 3 ) ^ { 2 } ( x + A )$$ where \(A\) is a constant to be found.
3. Sketch the graph of \(C\). Show clearly the coordinates of the points where \(C\) cuts or meets the \(x\)-axis and where \(C\) cuts the \(y\)-axis.

10 The diagram below shows the curve \(y = f ( x )\).
\includegraphics[max width=\textwidth, alt={}, center]{60e1e785-c34b-48ef-a63f-13a25fee186e-07_942_679_1500_242} Sketch the graph of the gradient function, \(y = f ^ { \prime } ( x )\), on the copy of the diagram in the Printed Answer Booklet.

1. The graph of \(y = f ( x )\) (where \(- 2 \leq x \leq 6\) ) has the following features:

• A local maximum at \(x = 0\).
• A local minimum at \(x = 2\).
• No other turning points.
• Three stationary points.

Sketch a possible graph of \(y = f ^ { \prime } ( x )\) on the axes provided.
You can ignore the scale for the \(y\)-axis.
\includegraphics[max width=\textwidth, alt={}, center]{4c649001-3816-4cfa-9418-e3e427df1eb5-04_1269_1354_826_447}
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9 A continuous curve has equation \(y = \mathrm { f } ( x )\) The curve passes through the points \(A ( 2,1 ) , B ( 4,5 )\) and \(C ( 6,1 )\)
It is given that \(f ^ { \prime } ( 4 ) = 0\)
Jasmin made two statements about the nature of the curve \(y = \mathrm { f } ( x )\) at the point \(B\) :
Statement 1: There is a turning point at \(B\)
Statement 2: There is a maximum point at \(B\)
9

1. Draw a sketch of the curve \(y = \mathrm { f } ( x )\) such that Statement 1 is correct and Statement 2 is correct.
   [0pt] [1 mark]
   \includegraphics[max width=\textwidth, alt={}, center]{9cd7f38d-a2a1-4fd3-9ed9-cb389e8ee3b6-10_593_588_1043_817} 9
2. Draw a sketch of the curve \(y = \mathrm { f } ( x )\) such that Statement 1 is correct and Statement 2 is not correct.
   \includegraphics[max width=\textwidth, alt={}, center]{9cd7f38d-a2a1-4fd3-9ed9-cb389e8ee3b6-11_607_597_497_813} 9
3. Draw a sketch of the curve \(y = \mathrm { f } ( x )\) such that Statement 1 is not correct and Statement 2 is not correct.
   [0pt] [1 mark]
   \includegraphics[max width=\textwidth, alt={}, center]{9cd7f38d-a2a1-4fd3-9ed9-cb389e8ee3b6-11_605_597_1541_813} 1 Charlie buys a car for \(\pounds 18000\) on 1 January 2016.
   The value of the car decreases exponentially.
   The car has a value of \(\pounds 12000\) on 1 January 2018.

4 The graph of $$y = \mathrm { f } ( x )$$ where $$f ( x ) = a x ^ { 2 } + b x + c$$ is shown in Figure 1. \begin{figure}[h] \end{figure} Which of the following shows the graph of \(y = \mathrm { f } ^ { \prime } ( x )\) ? Tick \(( \checkmark )\) one box.
\includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-05_2272_437_429_557}
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\includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-05_117_117_1151_1133}
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3 The function f is concave and is represented by one of the graphs below. Identify the graph which represents f . Tick ( \(\checkmark\) ) one box.
\includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-03_709_561_632_191}
\includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-03_117_111_927_826}
\includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-03_716_570_630_1082} □
\includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-03_711_563_1503_191}
\includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-03_711_565_1503_1085}
\includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-03_117_113_1800_1717}