Find stationary points

1 Given that \(y = 16 x + x ^ { - 1 }\), find the two values of \(x\) for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\).
(5 marks)

5 The diagram shows part of a curve with a maximum point \(M\).
\includegraphics[max width=\textwidth, alt={}, center]{22f2da99-0878-48a6-a2b7-1ba339d3c7e4-06_472_791_358_630} The equation of the curve is $$y = 15 x ^ { \frac { 3 } { 2 } } - x ^ { \frac { 5 } { 2 } }$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Hence find the coordinates of the maximum point \(M\).
3. The point \(P ( 1,14 )\) lies on the curve. Show that the equation of the tangent to the curve at \(P\) is \(y = 20 x - 6\).
4. The tangents to the curve at the points \(P\) and \(M\) intersect at the point \(R\). Find the length of \(R M\).

5 The diagram shows part of a curve with a maximum point \(M\).
\includegraphics[max width=\textwidth, alt={}, center]{258f0400-6e3b-406c-9f86-acc9fff4e094-4_480_645_354_694} The curve is defined for \(x \geqslant 0\) by the equation $$y = 6 x - 2 x ^ { \frac { 3 } { 2 } }$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   (3 marks)
2. 1. Hence find the coordinates of the maximum point \(M\).
   2. Write down the equation of the normal to the curve at \(M\).
3. The point \(P \left( \frac { 9 } { 4 } , \frac { 27 } { 4 } \right)\) lies on the curve.
   1. Find an equation of the normal to the curve at the point \(P\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are positive integers.
   2. The normals to the curve at the points \(M\) and \(P\) intersect at the point \(R\). Find the coordinates of \(R\).
      \(6 \quad\) A curve \(C\), defined for \(0 \leqslant x \leqslant 2 \pi\) by the equation \(y = \sin x\), where \(x\) is in radians, is sketched below. The region bounded by the curve \(C\), the \(x\)-axis from 0 to 2 and the line \(x = 2\) is shaded.
      \includegraphics[max width=\textwidth, alt={}, center]{258f0400-6e3b-406c-9f86-acc9fff4e094-5_441_789_466_612}

7. On a journey, the average speed of a car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). For \(v \geq 5\), the cost per kilometre, \(C\) pence, of the journey is modelled by \(C = \frac { 160 } { v } + \frac { v ^ { 2 } } { 100 }\).
Using this model,

1. show, by calculus, that there is a value of \(v\) for which \(C\) has a stationary value, and find this value of \(v\).
2. Justify that this value of \(v\) gives a minimum value of \(C\).
3. Find the minimum value of \(C\) and hence find the minimum cost of a 250 km car journey.

3. Find the coordinates of the stationary point of the curve with equation $$y = x + \frac { 4 } { x ^ { 2 } } .$$

7. $$f ( x ) = 2 + 6 x ^ { 2 } - x ^ { 3 }$$

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.

6. A curve has the equation $$y = x ^ { 3 } + a x ^ { 2 } - 15 x + b$$ where \(a\) and \(b\) are constants. Given that the curve is stationary at the point \(( - 1,12 )\),

1. find the values of \(a\) and \(b\),
2. find the coordinates of the other stationary point of the curve.

8. The curve \(C\) has the equation $$y = 3 - x ^ { \frac { 1 } { 2 } } - 2 x ^ { - \frac { 1 } { 2 } } , \quad x > 0 .$$

1. Find the coordinates of the points where \(C\) crosses the \(x\)-axis.
2. Find the exact coordinates of the stationary point of \(C\).
3. Determine the nature of the stationary point.
4. Sketch the curve \(C\).

8. The curve \(C\) has equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = 3 \ln x + \frac { 1 } { x } , \quad x > 0$$ The point \(P\) is a stationary point on \(C\).

1. Calculate the \(x\)-coordinate of \(P\).
2. Show that the \(y\)-coordinate of \(P\) may be expressed in the form \(k - k \ln k\), where \(k\) is a constant to be found. The point \(Q\) on \(C\) has \(x\)-coordinate 1 .
3. Find an equation for the normal to \(C\) at \(Q\). The normal to \(C\) at \(Q\) meets \(C\) again at the point \(R\).
4. Show that the \(x\)-coordinate of \(R\)
   1. satisfies the equation \(6 \ln x + x + \frac { 2 } { x } - 3 = 0\),
   2. lies between 0.13 and 0.14 .

\(\mathbf { 1 }\) & \(\mathbf { 3 }\)
\hline \end{tabular} \end{center} A curve \(C\) has equation \(y = x ^ { 3 } - 3 x ^ { 2 }\). a) Find the stationary points of \(C\) and determine their nature.
b) Draw a sketch of \(C\), clearly indicating the stationary points and the points where the curve crosses the coordinate axes.
c) Without performing the integration, state whether \(\int _ { 0 } ^ { 3 } \left( x ^ { 3 } - 3 x ^ { 2 } \right) \mathrm { d } x\) is positive or
negative, giving a reason for your answer. In each of the two statements below, \(c\) and \(d\) are real numbers. One of the statements is true, while the other is false. $$\begin{aligned} & \text { A : } \quad ( 2 c - d ) ^ { 2 } = 4 c ^ { 2 } - d ^ { 2 } , \text { for all values of } c \text { and } d .
& \text { B : } \quad 8 c ^ { 3 } - d ^ { 3 } = ( 2 c - d ) \left( 4 c ^ { 2 } + 2 c d + d ^ { 2 } \right) , \text { for all values of } c \text { and } d . \end{aligned}$$ a) Identify the statement which is false. Show, by counter example, that this statement is in fact false.
b) Identify the statement which is true. Give a proof to show that this statement is in fact true. The value of a car, \(\pounds V\), may be modelled as a continuous variable. At time \(t\) years, the value of the car is given by \(V = A \mathrm { e } ^ { k t }\), where \(A\) and \(k\) are constants. When the car is new, it is worth \(\pounds 30000\). When the car is two years old, it is worth \(\pounds 20000\). Determine the value of the car when it is six years old, giving your answer correct to the nearest \(\pounds 100\). The curve \(C\) has equation \(y = 7 + 13 x - 2 x ^ { 2 }\). The point \(P\) lies on \(C\) and is such that the tangent to \(C\) at \(P\) has equation \(y = x + c\), where \(c\) is a constant. Find the coordinates of \(P\) and the value of \(c\). a) Solve \(2 \log _ { 10 } x = 1 + \log _ { 10 } 5 - \log _ { 10 } 2\).
b) Solve \(3 = 2 \mathrm { e } ^ { 0 \cdot 5 x }\).
c) Express \(4 ^ { x } - 10 \times 2 ^ { x }\) in terms of \(y\), where \(y = 2 ^ { x }\). Hence solve the equation \(4 ^ { x } - 10 \times 2 ^ { x } = - 16\). The coordinates of three points \(A , B , C\) are \(( 4,6 ) , ( - 3,5 )\) and \(( 5 , - 1 )\) respectively. a) Show that \(B \widehat { A C }\) is a right angle.
b) A circle passes through all three points \(A , B , C\). Determine the equation of the circle.

12. A curve \(C\) has equation \(y = - x ^ { 3 } + 12 x - 20\).

1. Find the coordinates of the stationary points of \(C\) and determine their nature.
2. Determine the range of values of \(x\) for which the curve is decreasing. Give your answer in set notation.

10 In this question you must show detailed reasoning. Fig. C2.2 indicates that the curve \(\mathrm { y } = \frac { 4 \mathrm { x } ( \pi - \mathrm { x } ) } { \pi ^ { 2 } } - \sin \mathrm { x }\) has a stationary point near \(x = 3\).

• Verify that the \(x\)-coordinate of this stationary point is between 2.6 and 2.7.
• Show that this stationary point is a maximum turning point.

5 In this question you must show detailed reasoning.

1. Find the coordinates of the two stationary points on the graph of \(y = 15 - x ^ { 2 } - \frac { 16 } { x ^ { 2 } }\).
2. Show that both these stationary points are maximum points.

1. The curve C is defined by the equations \(y = x - 4 \sqrt { x }\), \(x \geq 0\)
   a) Find \(\frac { d y } { d x }\)
   b) Find the coordinates of the turning point of \(C\)
   c) Find the coordinates of the two \(x\)-intercepts of \(C\).
2. The cubic polynomial \(f ( \mathrm { x } )\) is defined by \(f ( x ) = x ^ { 3 } + k x ^ { 2 } + 9 x - 20\)
   a) Given that \(( x - 5 )\) is a factor of \(f ( x )\), find the value of \(k\).
   b) Show clearly that there is only one real solution to the equations \(f ( x ) = 0\)
   c) Given also that the function \(g ( x )\) is defined as \(g ( x ) = \log _ { 2 } x\), \(x > 0\)

Solve \(f g ( x ) = 0\)
3)
\includegraphics[max width=\textwidth, alt={}, center]{a202ddae-5ecd-4803-9a05-33c37d1880cd-05_537_990_137_338} The diagram above shows part of the curve with equation \(y = k \sin \left( x + \frac { \pi } { 3 } \right)\)
The curve meets the y -axis at \(( 0 , \sqrt { 3 } )\) and the x -axis at \(( p , 0 )\) and \(( q , 0 )\)
a) Find the value of the constant \(k\)
b) Find the value of \(p\) and the value of \(q\).
4) On the axes provided, sketch the curve \(y = \tan \left( \frac { x } { 2 } \right) , - 2 \pi \leq x \leq 2 \pi\) Mark clearly the coordinates of any points the curves crosses the coordinate axes and the equations of any asymptotes.
\includegraphics[max width=\textwidth, alt={}, center]{a202ddae-5ecd-4803-9a05-33c37d1880cd-05_661_979_2131_568}
5)
\includegraphics[max width=\textwidth, alt={}, center]{a202ddae-5ecd-4803-9a05-33c37d1880cd-06_643_661_132_667} The diagram above shows the cross-section of a small shed.
The straight line \(A B\) is vertical and has length 2.12 m . The straight line \(A D\) is horizontal and has length 1.86 m . The curve \(B C\) is an arc of a circle with centre \(A\), and \(C D\) is a straight line.
Given that the size of angle BAC is 0.65 radians:
a) Find the size of angle CAD giving your answer in radians to 2 dp .
b) Find the area of the cross-section \(A B C D\)
c) Find the perimeter of the cross-section ABCD
6) Prove, from first principles, that if \(f ( x ) = 3 x ^ { 2 }\) then the derivative \(f ^ { \prime } ( x )\) is given by \(f ^ { \prime } ( x ) = 6 x\)
7) Given \(f ( x ) = x ^ { 2 } + 1 , \quad x < - 1\) Find \(f ^ { - 1 } ( x )\) stating its domain and range
8) The curve \(C\) has equation \(y = x ^ { 2 } ( x - 6 ) + \frac { 4 } { x } , \quad x > 0\) The points P and Q lie on C and have x -coordinates 1 and 2 respectively.
a) Show that the length of PQ is V 170 .
b) Show that the tangents to C at P and Q are parallel.
c) Find an equation for the normal to C at P , giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
a) Solve, for \(0 \leq x < 360 ^ { \circ }\), giving your answers to 1 decimal place. $$5 \sin 2 x = 2 \cos 2 x$$ b) Solve for \(0 \leq x \leq 4 \pi\) giving your answers in radians to 3 significant figures. $$4 \sin ^ { 2 } x = 6 - 9 \cos x$$

9. The total cost \(C\), in \(\pounds\), for a certain car journey, is modelled by $$C = \frac { 200 } { V } + \frac { 2 V } { 25 } , V > 30 ,$$ where \(V\) is the average speed in miles per hour.
a) Find the value of \(V\) for which \(C\) is stationary.

8. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\).
The curve \(C\) crosses the \(x\)-axis at the origin, \(O\), and at the points \(A\) and \(B\) as shown in Figure 5. Given that $$f ^ { \prime } ( x ) = k - 4 x - 3 x ^ { 2 }$$ where \(k\) is constant,

1. show that \(C\) has a point of inflection at \(x = - \frac { 2 } { 3 }\) Given also that the distance \(A B = 4 \sqrt { 2 }\)
2. find, showing your working, the integer value of \(k\).
   (5)

2.

1. Solve the equation \(x ^ { 4 } - 10 x ^ { 2 } + 25 = 0\).
2. Given that \(y = \frac { 2 } { 5 } x ^ { 5 } - \frac { 20 } { 3 } x ^ { 3 } + 50 x + 3\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
3. Hence find the number of stationary points on the curve \(y = \frac { 2 } { 5 } x ^ { 5 } - \frac { 20 } { 3 } x ^ { 3 } + 50 x + 3\).

12. A curve is defined for \(x \geq 0\) by the equation $$y = 6 x - 2 x ^ { \frac { 3 } { 2 } }$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   [0pt] [2 marks]
2. The curve has one stationary point. Find the coordinates of the stationary point and determine whether it is a maximum or minimum point. Fully justify your answer.
   [0pt] [3 marks]
   [0pt] [BLANK PAGE]

3. In this question you must show detailed algebraic reasoning. Find the coordinates of any stationary points on the curve below. $$y = ( 1 - 3 x ) ( 3 - x ) ^ { 3 }$$

12.
\includegraphics[max width=\textwidth, alt={}, center]{e0c7b6f3-9c28-48bd-a401-635efb2521e3-26_504_856_239_657} The figure above shows the curve \(C\) with equation $$f ( x ) = \frac { x + 4 } { \sqrt { x } } , x > 0 .$$ a) Determine the coordinates of the minimum point of \(C\), labelled as \(M\). The point \(N\) lies on the \(x\) axis so that \(M N\) is parallel to the \(y\) axis. The finite region \(R\) is bounded by \(C\), the \(x\) axis, the straight line segment \(M N\) and the straight line with equation \(x = 1\).
b) Use the trapezium rule with 4 strips of equal width to estimate the area of \(R\).
[0pt] [BLANK PAGE]

17. \begin{figure}[h] \end{figure} \section*{In this question you must show detailed reasoning. Solutions relying entirely on calculator technology are not acceptable.} Figure 3 shows a sketch of part of the curve \(C\) with equation $$y = \frac { 2 } { 3 } x ^ { 2 } - 9 \sqrt { x } + 13 \quad x \geq 0$$

1. Find, using calculus, the range of values of \(x\) for which \(y\) is increasing. The point \(P\) lies on \(C\) and has coordinates (9, 40).
   The line \(l\) is the tangent to \(C\) at the point \(P\).
   The finite region \(R\), shown shaded in Figure 3, is bounded by the curve \(C\), the line \(l\), the \(x\)-axis and the \(y\)-axis.
2. Find, using calculus, the exact area of \(R\).
   (6) ADDITIONAL SHEET ADDITIONAL SHEET ADDITIONAL SHEET

5. A curve has the following properties:

• The gradient of the curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - 2 x\).
• The curve passes through the point \(( 4 , - 13 )\).

Determine the coordinates of the points where the curve meets the line \(y = 2 x\).
[0pt] [BLANK PAGE]

3 The function \(w = \mathrm { f } ( x , y , z )\) is given by \(\mathrm { f } ( x , y , z ) = x ^ { 2 } y z + 2 x y ^ { 2 } z + 3 x y z ^ { 2 } - 24 x y z\), for \(x , y , z \neq 0\).

1. (a) Find
   • \(\mathrm { f } _ { x }\),
   • \(\mathrm { f } _ { y }\),
   • \(\mathrm { f } _ { z }\).
     (b) Hence find the values of \(a , b , c\) and \(d\) for which \(w\) has a stationary value when \(d = \mathrm { f } ( a , b , c )\).
   • You are given that this stationary value is a local minimum of \(w\). Find values of \(x , y\) and \(z\) which show that it is not a global minimum of \(w\).