Find stationary points

10

1. Find the coordinates of the stationary points of the curve \(y = 2 x ^ { 3 } + 5 x ^ { 2 } - 4 x\).
2. State the set of values for \(x\) for which \(2 x ^ { 3 } + 5 x ^ { 2 } - 4 x\) is a decreasing function.
3. Show that the equation of the tangent to the curve at the point where \(x = \frac { 1 } { 2 }\) is \(10 x - 4 y - 7 = 0\).
4. Hence, with the aid of a sketch, show that the equation \(2 x ^ { 3 } + 5 x ^ { 2 } - 4 x = \frac { 5 } { 2 } x - \frac { 7 } { 4 }\) has two distinct real roots.

8

1. Find the coordinates of the stationary point on the curve \(y = 3 x ^ { 2 } - \frac { 6 } { x } - 2\).
2. Determine whether the stationary point is a maximum point or a minimum point.

8

1. Find the coordinates of the stationary point on the curve \(y = x ^ { 4 } + 32 x\).
2. Determine whether this stationary point is a maximum or a minimum.
3. For what values of \(x\) does \(x ^ { 4 } + 32 x\) increase as \(x\) increases?

10 The curve \(y = ( 1 - x ) \left( x ^ { 2 } + 4 x + k \right)\) has a stationary point when \(x = - 3\).

1. Find the value of the constant \(k\).
2. Determine whether the stationary point is a maximum or minimum point.
3. Given that \(y = 9 x - 9\) is the equation of the tangent to the curve at the point \(A\), find the coordinates of \(A\).

8 A curve has equation \(y = 3 x ^ { 3 } - 7 x + \frac { 2 } { x }\).

1. Verify that the curve has a stationary point when \(x = 1\).
2. Determine the nature of this stationary point.
3. The tangent to the curve at this stationary point meets the \(y\)-axis at the point \(Q\). Find the coordinates of \(Q\).

9 The curve \(y = 2 x ^ { 3 } - a x ^ { 2 } + 8 x + 2\) passes through the point \(B\) where \(x = 4\).

1. Given that \(B\) is a stationary point of the curve, find the value of the constant \(a\).
2. Determine whether the stationary point \(B\) is a maximum point or a minimum point.
3. Find the \(x\)-coordinate of the other stationary point of the curve.

11 The curve \(y = 4 x ^ { 2 } + \frac { a } { x } + 5\) has a stationary point. Find the value of the positive constant \(a\) given that the \(y\)-coordinate of the stationary point is 32 .

7 Differentiate \(4 x ^ { 2 } + \frac { 1 } { x }\) and hence find the \(x\)-coordinate of the stationary point of the curve \(y = 4 x ^ { 2 } + \frac { 1 } { x }\).

10

1. Differentiate \(x ^ { 3 } - 3 x ^ { 2 } - 9 x\). Hence find the \(x\)-coordinates of the stationary points on the curve \(y = x ^ { 3 } - 3 x ^ { 2 } - 9 x\), showing which is the maximum and which the minimum.
2. Find, in exact form, the coordinates of the points at which the curve crosses the \(x\)-axis.
3. Sketch the curve.

10

1. Use calculus to find, correct to 1 decimal place, the coordinates of the turning points of the curve \(y = x ^ { 3 } - 5 x\). [You need not determine the nature of the turning points.]
2. Find the coordinates of the points where the curve \(y = x ^ { 3 } - 5 x\) meets the axes and sketch the curve.
3. Find the equation of the tangent to the curve \(y = x ^ { 3 } - 5 x\) at the point \(( 1 , - 4 )\). Show that, where this tangent meets the curve again, the \(x\)-coordinate satisfies the equation $$x ^ { 3 } - 3 x + 2 = 0$$ Hence find the \(x\)-coordinate of the point where this tangent meets the curve again.

11. \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 \(k\).

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

The curve

• passes through the point \(P ( 3 , - 10 )\)
• has a turning point at \(P\)

Given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 x ^ { 3 } - 9 x ^ { 2 } + 5 x + k$$ where \(k\) is a constant,

1. show that \(k = 12\)
2. Hence find the coordinates of the point where \(C\) crosses the \(y\)-axis.

1. In this question you should show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve \(C\) with equation $$y = x ^ { 3 } - 10 x ^ { 2 } + 27 x - 23$$ The point \(P ( 5 , - 13 )\) lies on \(C\)
The line \(l\) is the tangent to \(C\) at \(P\)

1. Use differentiation to find the equation of \(l\), giving your answer in the form \(y = m x + c\) where \(m\) and \(c\) are integers to be found.
2. Hence verify that \(l\) meets \(C\) again on the \(y\)-axis. The finite region \(R\), shown shaded in Figure 2, is bounded by the curve \(C\) and the line \(l\).
3. Use algebraic integration to find the exact area of \(R\).

4

1. Find the coordinates of the stationary points on the curve \(y = x ^ { 3 } - 6 x ^ { 2 } + 9 x\).
2. The equation \(x ^ { 3 } - 6 x ^ { 2 } + 9 x + k = 0\) has exactly one real root. Using your answers from part (a) or otherwise, find the range of possible values of \(k\).

5 The fuel consumption of a car, \(C\) miles per gallon, varies with the speed, \(v\) miles per hour. Jamal models the fuel consumption of his car by the formula
\(C = \frac { 12 } { 5 } v - \frac { 3 } { 125 } v ^ { 2 }\), for \(0 \leqslant v \leqslant 80\).

1. Suggest a reason why Jamal has included an upper limit in his model.
2. Determine the speed that gives the maximum fuel consumption. Amaya's car does more miles per gallon than Jamal's car. She proposes to model the fuel consumption of her car using a formula of the form
   \(C = \frac { 12 } { 5 } v - \frac { 3 } { 125 } v ^ { 2 } + k\), for \(0 \leqslant v \leqslant 80\), where \(k\) is a positive constant.
3. Give a reason why this model is not suitable.
4. Suggest a different change to Jamal's formula which would give a more suitable model.

6

1. Determine the two real roots of the equation \(8 x ^ { 6 } + 7 x ^ { 3 } - 1 = 0\).
2. Determine the coordinates of the stationary points on the curve \(y = 8 x ^ { 7 } + \frac { 49 } { 4 } x ^ { 4 } - 7 x\).
3. For each of the stationary points, use the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) to determine whether it is a maximum or a minimum.

9 The equation of a curve is \(y = 12 x - 4 x ^ { \frac { 3 } { 2 } }\).

1. State the coordinates of the intersection of the curve with the \(y\)-axis.
2. Find the value of \(y\) when \(x = 9\).
3. Determine the coordinates of the stationary point.
4. Sketch the curve, giving the coordinates of the stationary point and of any intercepts with the axes.

11 In this question you must show detailed reasoning.
The equation of a curve is \(y = 2 x ^ { 3 } + 9 x ^ { 2 } + 24 x - 8\).
Show that there are no stationary points on this curve.

16 The equation of a curve is
\(y = 6 x ^ { 4 } + 8 x ^ { 3 } - 21 x ^ { 2 } + 12 x - 6\).

1. In this question you must show detailed reasoning. Determine
   • The coordinates of the stationary points on the curve.
   • The nature of the stationary points on the curve.
   • The \(x\)-coordinate of the non-stationary point of inflection on the curve.
   • On the axes in the Printed Answer Booklet, sketch the curve whose equation is
   $$y = 6 x ^ { 4 } + 8 x ^ { 3 } - 21 x ^ { 2 } + 12 x - 6 .$$

8 The equation of a curve is \(y = 2 x ^ { 3 } + 3 m x ^ { 2 } - 9 m x + 4\). Determine the range of values of \(m\) for which the curve has no stationary values.

2 The surface with equation \(z = 6 x ^ { 3 } + \frac { 1 } { 9 } y ^ { 2 } + x ^ { 2 } y\) has two stationary points.

1. Verify that one of these stationary points is at the origin.
2. Find the coordinates of the second stationary point.

2 The surface \(S\) has equation \(z = x ^ { 3 } + y ^ { 3 } - 2 x ^ { 2 } - 5 y ^ { 2 } + 3 x y\).
It is given that \(S\) has two stationary points; one at the origin, \(O\), and the other at the point \(A\).
Determine the coordinates of \(A\).

3 A surface has equation \(z = x ^ { 2 } y ^ { 2 } - 3 x y + 2 x + y\) for all real values of \(x\) and \(y\). Determine the coordinates of all stationary points of this surface.

1 The curve with equation \(y = 13 + 18 x + 3 x ^ { 2 } - 4 x ^ { 3 }\) passes through the point \(P\) where \(x = - 1\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Show that the point \(P\) is a stationary point of the curve and find the other value of \(x\) where the curve has a stationary point.
3. 1. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at the point \(P\).
   2. Hence, or otherwise, determine whether \(P\) is a maximum point or a minimum point.
      (l mark)

5 The curve with equation \(y = x ^ { 3 } - 10 x ^ { 2 } + 28 x\) is sketched below.
\includegraphics[max width=\textwidth, alt={}, center]{f2c95d73-d3fe-48f7-af07-84f12bb06727-3_483_899_402_568} The curve crosses the \(x\)-axis at the origin \(O\) and the point \(A ( 3,21 )\) lies on the curve.

1. 1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   2. Hence show that the curve has a stationary point when \(x = 2\) and find the \(x\)-coordinate of the other stationary point.
2. 1. Find \(\int \left( x ^ { 3 } - 10 x ^ { 2 } + 28 x \right) \mathrm { d } x\).
   2. Hence show that \(\int _ { 0 } ^ { 3 } \left( x ^ { 3 } - 10 x ^ { 2 } + 28 x \right) \mathrm { d } x = 56 \frac { 1 } { 4 }\).
   3. Hence determine the area of the shaded region bounded by the curve and the line \(O A\).