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 gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 4 x + 3\). The curve passes through the point ( 2,9 ).

1. Find the equation of the tangent to the curve at the point \(( 2,9 )\).
2. Find the equation of the curve and the coordinates of its points of intersection with the \(x\)-axis. Find also the coordinates of the minimum point of this curve.
3. Find the equation of the curve after it has been stretched parallel to the \(x\)-axis with scale factor \(\frac { 1 } { 2 }\). Write down the coordinates of the minimum point of the transformed curve.

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.

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}
□ \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}

A curve with equation \(y = f(x)\) passes through the point \((3, 6)\). Given that $$f'(x) = (x - 2)(3x + 4)$$

1. use integration to find \(f(x)\). Give your answer as a polynomial in its simplest form. [5]
2. Show that \(f(x) = (x - 2)^2(x + p)\), where \(p\) is a positive constant. State the value of \(p\). [3]
3. Sketch the graph of \(y = f(x)\), showing the coordinates of any points where the curve touches or crosses the coordinate axes. [4]

\includegraphics{figure_1} Figure 1 shows the curve with equation \(y = \text{f}(x)\) which crosses the \(x\)-axis at the origin and at the points \(A\) and \(B\). Given that $$\text{f}'(x) = 6 - 4x - 3x^2,$$

1. find an expression for \(y\) in terms of \(x\), [5]
2. show that \(AB = k\sqrt{7}\), where \(k\) is an integer to be found. [6]