Find stationary points

2. In this question you must show all stages of your working. \section*{Solutions relying on calculator technology are not acceptable.} A curve has equation $$y = 3 x ^ { 5 } + 4 x ^ { 3 } - x + 5$$ The points \(P\) and \(Q\) lie on the curve.
The gradient of the curve at both point \(P\) and point \(Q\) is 2
Find the \(x\) coordinates of \(P\) and \(Q\).

9. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = \frac { 8 } { x } + \frac { 1 } { 2 } x - 5 , \quad 0 < x \leqslant 12$$ The curve crosses the \(x\)-axis at \(( 2,0 )\) and \(( 8,0 )\) and has a minimum point at \(A\).

1. Use calculus to find the coordinates of point \(A\).
2. State
   1. the roots of the equation \(2 \mathrm { f } ( x ) = 0\)
   2. the coordinates of the turning point on the curve \(y = \mathrm { f } ( x ) + 2\)
   3. the roots of the equation \(\mathrm { f } ( 4 x ) = 0\)

5. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve with equation $$y = 27 \sqrt { x } - 2 x ^ { 2 } , \quad x \in \mathbb { R } , x > 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) The curve has a maximum turning point \(P\), as shown in Figure 2.
2. Use the answer to part (a) to find the exact coordinates of \(P\).

5. A company makes a particular type of watch. The annual profit made by the company from sales of these watches is modelled by the equation $$P = 12 x - x ^ { \frac { 3 } { 2 } } - 120$$ where \(P\) is the annual profit measured in thousands of pounds and \(\pounds x\) is the selling price of the watch. According to this model,

1. find, using calculus, the maximum possible annual profit.
2. Justify, also using calculus, that the profit you have found is a maximum.

10. The curve \(C\) has equation $$y = a x ^ { 3 } - 3 x ^ { 2 } + 3 x + b$$ where \(a\) and \(b\) are constants. Given that

• the point \(( 2,5 )\) lies on \(C\)
• the gradient of the curve at \(( 2,5 )\) is 7
  1. find the value of \(a\) and the value of \(b\).
  2. Prove that \(C\) has no turning points.
     

8. In this question you must 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 $$y = \frac { 4 } { 3 } x ^ { 3 } - 11 x ^ { 2 } + k x \quad \text { where } k \text { is a constant }$$ The point \(M\) is the maximum turning point of \(C\) and is shown in Figure 2.
Given that the \(x\) coordinate of \(M\) is 2

1. show that \(k = 28\)
2. Determine the range of values of \(x\) for which \(y\) is increasing. The line \(l\) passes through \(M\) and is parallel to the \(x\)-axis.
   The region \(R\), shown shaded in Figure 2, is bounded by the curve \(C\), the line \(l\) and the \(y\)-axis.
3. Find, by algebraic integration, the exact area of \(R\).

9. The curve \(C\) has equation \(y = 12 \sqrt { } ( x ) - x ^ { \frac { 3 } { 2 } } - 10 , \quad x > 0\)

1. Use calculus to find the coordinates of the turning point on \(C\).
2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
3. State the nature of the turning point.

8. The curve \(C\) has equation \(y = 6 - 3 x - \frac { 4 } { x ^ { 3 } } , x \neq 0\)

1. Use calculus to show that the curve has a turning point \(P\) when \(x = \sqrt { } 2\)
2. Find the \(x\)-coordinate of the other turning point \(Q\) on the curve.
3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
4. Hence or otherwise, state with justification, the nature of each of these turning points \(P\) and \(Q\).

3. The curve \(C\) has equation $$y = 2 \sqrt { } x + \frac { 18 } { \sqrt { } x } - 1 , \quad x > 0$$

1. Find
   1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
   2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)
2. Use calculus to find the coordinates of the stationary point of \(C\).
3. Determine whether the stationary point is a maximum or minimum, giving a reason for your answer.
   \includegraphics[max width=\textwidth, alt={}, center]{e7043e7a-2c8f-425a-8471-f647828cc297-09_138_154_2597_1804}

Find the coordinates of the stationary point on the curve with equation \(y = 2 x ^ { 2 } - 12 x\).

10. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve with equation \(y = x ^ { 3 } - 8 x ^ { 2 } + 20 x\). The curve has stationary points \(A\) and \(B\).

1. Use calculus to find the \(x\)-coordinates of \(A\) and \(B\).
2. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at \(A\), and hence verify that \(A\) is a maximum. The line through \(B\) parallel to the \(y\)-axis meets the \(x\)-axis at the point \(N\).
   The region \(R\), shown shaded in Figure 3, is bounded by the curve, the \(x\)-axis and the line from \(A\) to \(N\).
3. Find \(\int \left( x ^ { 3 } - 8 x ^ { 2 } + 20 x \right) \mathrm { d } x\).
4. Hence calculate the exact area of \(R\).

1. Using calculus, find the coordinates of the stationary point on the curve with equation

$$y = 2 x + 3 + \frac { 8 } { x ^ { 2 } } , \quad x > 0$$

1. The curve with equation

$$y = x ^ { 2 } - 32 \sqrt { } ( x ) + 20 , \quad x > 0$$ has a stationary point \(P\). Use calculus

1. to find the coordinates of \(P\),
2. to determine the nature of the stationary point \(P\).

9. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve with equation $$y = 7 x ^ { 2 } ( 5 - 2 \sqrt { x } ) , \quad x \geqslant 0$$ The curve has a turning point at the point \(A\), where \(x > 0\), as shown in Figure 3.

1. Using calculus, find the coordinates of the point \(A\). The curve crosses the \(x\)-axis at the point \(B\), as shown in Figure 3.
2. Use algebra to find the \(x\) coordinate of the point \(B\). The finite region \(R\), shown shaded in Figure 3, is bounded by the curve, the line through \(A\) parallel to the \(x\)-axis and the line through \(B\) parallel to the \(y\)-axis.
3. Use integration to find the area of the region \(R\), giving your answer to 2 decimal places.
   

   END

5. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = ( 8 - x ) \ln x , \quad x > 0$$ The curve cuts the \(x\)-axis at the points \(A\) and \(B\) and has a maximum turning point at \(Q\), as shown in Figure 1.

1. Write down the coordinates of \(A\) and the coordinates of \(B\).
2. Find f'(x).
3. Show that the \(x\)-coordinate of \(Q\) lies between 3.5 and 3.6
4. Show that the \(x\)-coordinate of \(Q\) is the solution of $$x = \frac { 8 } { 1 + \ln x }$$ To find an approximation for the \(x\)-coordinate of \(Q\), the iteration formula $$x _ { n + 1 } = \frac { 8 } { 1 + \ln x _ { n } }$$ is used.
5. Taking \(x _ { 0 } = 3.55\), find the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\). Give your answers to 3 decimal places.

9

1. Find the gradient of the curve \(y = 2 x ^ { 2 }\) at the point where \(x = 3\).
2. At a point \(A\) on the curve \(y = 2 x ^ { 2 }\), the gradient of the normal is \(\frac { 1 } { 8 }\). Find the coordinates of \(A\). Points \(P _ { 1 } \left( 1 , y _ { 1 } \right) , P _ { 2 } \left( 1.01 , y _ { 2 } \right)\) and \(P _ { 3 } \left( 1.1 , y _ { 3 } \right)\) lie on the curve \(y = k x ^ { 2 }\). The gradient of the chord \(P _ { 1 } P _ { 3 }\) is 6.3 and the gradient of the chord \(P _ { 1 } P _ { 2 }\) is 6.03.
3. What do these results suggest about the gradient of the tangent to the curve \(y = k x ^ { 2 }\) at \(P _ { 1 }\) ?
4. Deduce the value of \(k\).

6

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

8

1. Find the coordinates of the stationary points of the curve \(y = 27 + 9 x - 3 x ^ { 2 } - x ^ { 3 }\).
2. Determine, in each case, whether the stationary point is a maximum or minimum point.
3. Hence state the set of values of \(x\) for which \(27 + 9 x - 3 x ^ { 2 } - x ^ { 3 }\) is an increasing function.
   \(9 \quad A\) is the point \(( 2,7 )\) and \(B\) is the point \(( - 1 , - 2 )\).

8

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

10

1. Given that \(y = \frac { 1 } { 3 } x ^ { 3 } - 9 x\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Find the coordinates of the stationary points on the curve \(y = \frac { 1 } { 3 } x ^ { 3 } - 9 x\).
3. Determine whether each stationary point is a maximum point or a minimum point.
4. Given that \(24 x + 3 y + 2 = 0\) is the equation of the tangent to the curve at the point ( \(p , q\) ), find \(p\) and \(q\).

8 The curve \(y = x ^ { 3 } - k x ^ { 2 } + x - 3\) has two stationary points.

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Given that there is a stationary point when \(x = 1\), find the value of \(k\).
3. Determine whether this stationary point is a minimum or maximum point.
4. Find the \(x\)-coordinate of the other stationary point.

8

1. Find the coordinates of the stationary points on the curve \(y = 2 x ^ { 3 } - 3 x ^ { 2 } - 12 x - 7\).
2. Determine whether each stationary point is a maximum point or a minimum point.
3. By expanding the right-hand side, show that $$2 x ^ { 3 } - 3 x ^ { 2 } - 12 x - 7 = ( x + 1 ) ^ { 2 } ( 2 x - 7 )$$
4. Sketch the curve \(y = 2 x ^ { 3 } - 3 x ^ { 2 } - 12 x - 7\), marking the coordinates of the stationary points and the points where the curve meets the axes.

6. The diagram shows the curve with equation \(y = 3 x - x ^ { \frac { 3 } { 2 } } , x \geq 0\). The curve meets the \(x\)-axis at the origin and at the point \(A\) and has a maximum at the point \(B\).

1. Find the \(x\)-coordinate of \(A\).
2. Find the coordinates of \(B\).

7. $$f ( x ) = x ^ { 3 } - 9 x ^ { 2 }$$

1. Find \(\mathrm { f } ^ { \prime } ( x )\).
2. Find \(\mathrm { f } ^ { \prime \prime } ( x )\).
3. Find the coordinates of the stationary points of the curve \(y = \mathrm { f } ( x )\).
4. Determine whether each stationary point is a maximum or a minimum point.