Find stationary points of polynomial

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

$$f ( x ) = ( x - 4 ) ( 2 x + 1 ) ^ { 2 }$$ The curve touches the \(x\)-axis at the point \(P\) and crosses the \(x\)-axis at the point \(Q\).

1. State the coordinates of the point \(P\).
2. Find \(f ^ { \prime } ( x )\).
3. Hence show that the equation of the tangent to the curve at the point where \(x = \frac { 5 } { 2 }\) can be expressed in the form \(y = k\), where \(k\) is a constant to be found. The curve with equation \(y = \mathrm { f } ( x + a )\), where \(a\) is a constant, passes through the origin \(O\).
4. State the possible values of \(a\).
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4. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = x ^ { 2 } + \frac { 16 } { x } , \quad x > 0$$ The curve has a minimum turning point at \(A\).

1. Find \(\mathrm { f } ^ { \prime } ( x )\).
2. Hence find the coordinates of \(A\).
3. Use your answer to part (b) to write down the turning point of the curve with equation
   1. \(y = \mathrm { f } ( x + 1 )\),
   2. \(y = \frac { 1 } { 2 } \mathrm { f } ( x )\).

12. \begin{figure}[h] \end{figure} Diagram not drawn to scale Figure 2 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = \frac { x ^ { 3 } - 9 x ^ { 2 } - 81 x } { 27 }$$ The curve crosses the \(x\)-axis at the point \(A\), the point \(B\) and the origin \(O\). The curve has a maximum turning point at \(C\) and a minimum turning point at \(D\).

1. Use algebra to find exact values for the \(x\) coordinates of the points \(A\) and \(B\).
2. Use calculus to find the coordinates of the points \(C\) and \(D\). The graph of \(y = \mathrm { f } ( x + a )\), where \(a\) is a constant, has its minimum turning point on the \(y\)-axis.
3. Write down the value of \(a\).
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3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = ( 2 x - 5 ) \mathrm { e } ^ { x } , \quad x \in \mathbb { R }$$ The curve has a minimum turning point at \(A\).

1. Use calculus to find the exact coordinates of \(A\). Given that the equation \(\mathrm { f } ( x ) = k\), where \(k\) is a constant, has exactly two roots,
2. state the range of possible values of \(k\).
3. Sketch the curve with equation \(y = | \mathrm { f } ( x ) |\). Indicate clearly on your sketch the coordinates of the points at which the curve crosses or meets the axes.

\begin{enumerate} \setcounter{enumi}{7} \item \(f ( x ) = 2 + 6 x ^ { 2 } - x ^ { 3 }\).

1. Find the coordinates of the stationary points of the curve \(y = \mathrm { f } ( x )\).
2. Determine whether each stationary point is a maximum or minimum point.
3. Sketch the curve \(y = \mathrm { f } ( x )\).
4. State the set of values of \(k\) for which the equation \(\mathrm { f } ( x ) = k\) has three solutions. \item The points \(P\) and \(Q\) have coordinates \(( 7,4 )\) and \(( 9,7 )\) respectively.

10. The curve with equation \(y = ( 2 - x ) ( 3 - x ) ^ { 2 }\) crosses the \(x\)-axis at the point \(A\) and touches the \(x\)-axis at the point \(B\).

1. Sketch the curve, showing the coordinates of \(A\) and \(B\).
2. Show that the tangent to the curve at \(A\) has the equation $$x + y = 2$$ Given that the curve is stationary at the points \(B\) and \(C\),
3. find the exact coordinates of \(C\).

9 The equation of a curve is given by \(y = ( x - 1 ) ^ { 2 } ( x + 2 )\).

1. Write \(( x - 1 ) ^ { 2 } ( x + 2 )\) in the form \(x ^ { 3 } + p x ^ { 2 } + q x + r\) where \(p , q\) and \(r\) are to be determined.
2. Show that the curve \(y = ( x - 1 ) ^ { 2 } ( x + 2 )\) has a maximum point when \(x = - 1\) and find the coordinates of the minimum point.
3. Sketch the curve \(y = ( x - 1 ) ^ { 2 } ( x + 2 )\).
4. For what values of \(k\) does \(( x - 1 ) ^ { 2 } ( x + 2 ) = k\) have exactly one root.

7 A curve has equation \(y = ( x + 2 ) \left( x ^ { 2 } - 3 x + 5 \right)\).

1. Find the coordinates of the minimum point, justifying that it is a minimum.
2. Calculate the discriminant of \(x ^ { 2 } - 3 x + 5\).
3. Explain why \(( x + 2 ) \left( x ^ { 2 } - 3 x + 5 \right)\) is always positive for \(x > - 2\).

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

$$f ( x ) = a x ^ { 3 } + 15 x ^ { 2 } - 39 x + b$$ and \(a\) and \(b\) are constants.
Given

• the point \(( 2,10 )\) lies on \(C\)
• the gradient of the curve at \(( 2,10 )\) is - 3
  1. (i) show that the value of \(a\) is - 2
     (ii) find the value of \(b\).
  2. Hence show that \(C\) has no stationary points.
  3. Write \(\mathrm { f } ( x )\) in the form \(( x - 4 ) \mathrm { Q } ( x )\) where \(\mathrm { Q } ( x )\) is a quadratic expression to be found.
  4. Hence deduce the coordinates of the points of intersection of the curve with equation

$$y = \mathrm { f } ( 0.2 x )$$ and the coordinate axes.

8. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve \(C\) with the equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \left( 2 x ^ { 2 } - 9 x + 9 \right) e ^ { - x } , \quad x \in R$$ The curve has a minimum turning point at \(A\) and a maximum turning point at \(B\) as shown in the figure above.
a. Find the coordinates of the point where \(C\) crosses the \(y\)-axis.
b. Show that \(\mathrm { f } ^ { \prime } ( x ) = - \left( 2 x ^ { 2 } - 13 x + 18 \right) e ^ { - x }\)
c. Hence find the exact coordinates of the turning points of \(C\). The graph with equation \(y = \mathrm { f } ( x )\) is transformed onto the graph with equation $$y = a \mathrm { f } ( x ) + b , \quad x \geq 0$$ The range of the graph with equation \(y = a \mathrm { f } ( x ) + b\) is \(0 \leq y \leq 9 e ^ { 2 } + 1\)
Given that \(a\) and \(b\) are constants.
d. find the value of \(a\) and the value of \(b\).

6. \begin{figure}[h] \end{figure} The figure 1 shows sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\). $$f ( x ) = a x ( x - b ) ^ { 2 } , x \in R$$ where \(a\) and \(b\) are constants.
The curve passes through the origin and touches the \(x\)-axis at the point \(( 3,0 )\).
There is a minimum point at \(( 1 , - 4 )\) and a maximum point at \(( 3,0 )\).
a. Find the equation of \(C\).
b. Deduce the values of \(x\) for which $$\mathrm { f } ^ { \prime } ( x ) > 0$$ Given that the line with equation \(y = k\), where \(k\) is a constant, intersects \(C\) at exactly one point,
c. State the possible values for \(k\).

6. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of a curve \(C\) with equation \(y = \mathrm { f } ( x )\) where \(\mathrm { f } ( x )\) is a cubic expression in \(X\). The curve

• passes through the origin
• has a maximum turning point at \(( 2,8 )\)
• has a minimum turning point at \(( 6,0 )\)
  1. Write down the set of values of \(x\) for which

$$\mathrm { f } ^ { \prime } ( x ) < 0$$ The line with equation \(y = k\), where \(k\) is a constant, intersects \(C\) at only one point.

Find the set of values of \(k\), giving your answer in set notation.

Find the equation of \(C\). You may leave your answer in factorised form.

8. \begin{figure}[h] \end{figure} Figure 2 shows part of the curve with equation $$y = x ^ { 3 } - 6 x ^ { 2 } + 9 x .$$ The curve touches the \(x\)-axis at \(A\) and has a maximum turning point at \(B\).

1. Show that the equation of the curve may be written as $$y = x ( x - 3 ) ^ { 2 } ,$$ and hence write down the coordinates of \(A\).
2. Find the coordinates of \(B\). The shaded region \(R\) is bounded by the curve and the \(x\)-axis.
3. Find the area of \(R\).

3. Given that \(\mathrm { f } ( x ) = 15 - 7 x - 2 x ^ { 2 }\),

1. find the coordinates of all points at which the graph of \(y = \mathrm { f } ( x )\) crosses the coordinate axes.
2. Sketch the graph of \(y = \mathrm { f } ( x )\).
3. Calculate the coordinates of the stationary point of \(\mathrm { f } ( x )\).
   [0pt] [P1 June 2002 Question 3]

9. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of a curve \(C\) with equation \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x )\) is a quartic expression in \(x\). The curve

• has maximum turning points at \(( - 1,0 )\) and \(( 5,0 )\)
• crosses the \(y\)-axis at \(( 0 , - 75 )\)
• has a minimum turning point at \(x = 2\)
  1. Find the set of values of \(x\) for which

$$\mathrm { f } ^ { \prime } ( x ) \geqslant 0$$ writing your answer in set notation.

Find the equation of \(C\). You may leave your answer in factorised form. The curve \(C _ { 1 }\) has equation \(y = \mathrm { f } ( x ) + k\), where \(k\) is a constant.
Given that the graph of \(C _ { 1 }\) intersects the \(x\)-axis at exactly four places,

find the range of possible values for \(k\).
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2 In this question you must show detailed reasoning. Find the values of \(x\) for which the gradient of the curve \(y = \frac { 2 } { 3 } x ^ { 3 } + \frac { 5 } { 2 } x ^ { 2 } - 3 x + 7\) is positive. Give your answer in set notation.

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1. Sketch the graph of any cubic function that has both three distinct real roots and a positive coefficient of \(x ^ { 3 }\)
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2. The function \(\mathrm { f } ( x )\) is defined by $$\mathrm { f } ( x ) = x ^ { 3 } + 3 p x ^ { 2 } + q$$ where \(p\) and \(q\) are constants and \(p > 0\)
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3. 1. Show that there is a turning point where the curve crosses the \(y\)-axis.
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4. (ii) The equation \(\mathrm { f } ( x ) = 0\) has three distinct real roots. By considering the positions of the turning points find, in terms of \(p\), the range of possible values of \(q\).