Find stationary points and nature

9 The equation of a curve is \(y = 3 x + 1 - 4 ( 3 x + 1 ) ^ { \frac { 1 } { 2 } }\) for \(x > - \frac { 1 } { 3 }\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
2. Find the coordinates of the stationary point of the curve and determine its nature.

10 \includegraphics[max width=\textwidth, alt={}, center]{51bd3ba6-e1d1-4c07-81cd-d99dd77f9306-14_832_830_276_653} The diagram shows the points \(A \left( 1 \frac { 1 } { 2 } , 5 \frac { 1 } { 2 } \right)\) and \(B \left( 7 \frac { 1 } { 2 } , 3 \frac { 1 } { 2 } \right)\) lying on the curve with equation \(y = 9 x - ( 2 x + 1 ) ^ { \frac { 3 } { 2 } }\).

1. Find the coordinates of the maximum point of the curve.
2. Verify that the line \(A B\) is the normal to the curve at \(A\).
3. Find the area of the shaded region.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

10 \includegraphics[max width=\textwidth, alt={}, center]{cbcb15b4-1870-4dfd-b6e9-839aa4601511-4_515_885_662_630} The diagram shows the curve \(y = x ^ { 3 } - 3 x ^ { 2 } - 9 x + k\), where \(k\) is a constant. The curve has a minimum point on the \(x\)-axis.

1. Find the value of \(k\).
2. Find the coordinates of the maximum point of the curve.
3. State the set of values of \(x\) for which \(x ^ { 3 } - 3 x ^ { 2 } - 9 x + k\) is a decreasing function of \(x\).
4. Find the area of the shaded region.

10 The equation of a curve is \(y = 2 x + \frac { 8 } { x ^ { 2 } }\).

1. Obtain expressions for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
2. Find the coordinates of the stationary point on the curve and determine the nature of the stationary point.
3. Show that the normal to the curve at the point \(( - 2 , - 2 )\) intersects the \(x\)-axis at the point \(( - 10,0 )\).
4. Find the area of the region enclosed by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 2\).

9 A curve has equation \(y = \mathrm { f } ( x )\) and is such that \(\mathrm { f } ^ { \prime } ( x ) = 3 x ^ { \frac { 1 } { 2 } } + 3 x ^ { - \frac { 1 } { 2 } } - 10\).

1. By using the substitution \(u = x ^ { \frac { 1 } { 2 } }\), or otherwise, find the values of \(x\) for which the curve \(y = \mathrm { f } ( x )\) has stationary points.
2. Find \(\mathrm { f } ^ { \prime \prime } ( x )\) and hence, or otherwise, determine the nature of each stationary point.
3. It is given that the curve \(y = \mathrm { f } ( x )\) passes through the point \(( 4 , - 7 )\). Find \(\mathrm { f } ( x )\).

5 A curve has equation \(y = 8 x + ( 2 x - 1 ) ^ { - 1 }\). Find the values of \(x\) at which the curve has a stationary point and determine the nature of each stationary point, justifying your answers.

9 The equation of a curve is \(y = 8 \sqrt { } x - 2 x\).

1. Find the coordinates of the stationary point of the curve.
2. Find an expression for \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and hence, or otherwise, determine the nature of the stationary point.
3. Find the values of \(x\) at which the line \(y = 6\) meets the curve.
4. State the set of values of \(k\) for which the line \(y = k\) does not meet the curve.

8 A curve has equation \(y = x ^ { 3 } + 3 x ^ { 2 } - 9 x + k\), where \(k\) is a constant.

1. Write down an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Find the \(x\)-coordinates of the two stationary points on the curve.
3. Hence find the two values of \(k\) for which the curve has a stationary point on the \(x\)-axis.

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

1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) in terms of \(x\).
2. Find the \(x\)-coordinates of the two stationary points and determine the nature of each stationary point.

8 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - x ^ { 2 } + 5 x - 4\).

1. Find the \(x\)-coordinate of each of the stationary points of the curve.
2. Obtain an expression for \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and hence or otherwise find the nature of each of the stationary points.
3. Given that the curve passes through the point \(( 6,2 )\), find the equation of the curve.

5 \includegraphics[max width=\textwidth, alt={}, center]{c703565b-8aa8-424b-9684-6592d4effdf8-2_554_689_1354_726} The diagram shows part of the curve $$y = 2 \cos x - \cos 2 x$$ and its maximum point \(M\). The shaded region is bounded by the curve, the axes and the line through \(M\) parallel to the \(y\)-axis.

1. Find the exact value of the \(x\)-coordinate of \(M\).
2. Find the exact value of the area of the shaded region.

3 The equation of a curve is \(y = \tan \frac { 1 } { 2 } x + 3 \sin \frac { 1 } { 2 } x\). The curve has a stationary point \(M\) in the interval \(\pi < x < 2 \pi\). Find the coordinates of \(M\), giving each coordinate correct to 3 significant figures.

3 \includegraphics[max width=\textwidth, alt={}, center]{cc7e798e-0817-405c-bae0-b24b9f451fbf-04_378_486_260_826} The diagram shows the curve with equation $$y = 5 \sin 2 x - 3 \tan 2 x$$ for values of \(x\) such that \(0 \leqslant x < \frac { 1 } { 4 } \pi\). Find the \(x\)-coordinate of the stationary point \(M\), giving your answer correct to 3 significant figures.

1. (Solutions based entirely on graphical or numerical methods are not acceptable.)

Given $$\mathrm { f } ( x ) = 2 x ^ { \frac { 5 } { 2 } } - 40 x + 8 \quad x > 0$$

1. solve the equation \(\mathrm { f } ^ { \prime } ( x ) = 0\)
2. solve the equation \(\mathrm { f } ^ { \prime \prime } ( x ) = 5\)

7. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. $$f ( x ) = 2 x - 3 \sqrt { x } - 5 \quad x > 0$$

1. Solve the equation $$f ( x ) = 9$$
2. Solve the equation $$\mathrm { f } ^ { \prime \prime } ( x ) = 6$$

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

$$f ( x ) = 2 x ^ { 3 } - k x ^ { 2 } + 14 x + 24$$ and \(k\) is a constant.

1. Find, in simplest form,
   1. \(\mathrm { f } ^ { \prime } ( x )\)
   2. \(\mathrm { f } ^ { \prime \prime } ( x )\) The curve with equation \(y = \mathrm { f } ^ { \prime } ( x )\) intersects the curve with equation \(y = \mathrm { f } ^ { \prime \prime } ( x )\) at the points \(A\) and \(B\). Given that the \(x\) coordinate of \(A\) is 5
2. find the value of \(k\).
3. Hence find the coordinates of \(B\).

1. In this question you must show detailed reasoning.

Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve with equation $$y = \frac { 1 } { 2 } x ^ { 2 } + \frac { 1458 } { \sqrt { x ^ { 3 } } } - 74 \quad x > 0$$ The point \(P\) is the only stationary point on the curve.

1. Use calculus to show that the \(x\) coordinate of \(P\) is 9 The line \(l\) passes through the point \(P\) and is parallel to the \(x\)-axis.
   The region \(R\), shown shaded in Figure 2, is bounded by the curve, the line \(l\) and the line with equation \(x = 4\)
2. Use algebraic integration to find the exact area of \(R\).

1. The curve \(C\) has equation

$$y = \frac { 12 x ^ { 3 } ( x - 7 ) + 14 x ( 13 x - 15 ) } { 21 \sqrt { x } } \quad x > 0$$

1. Write the equation of \(C\) in the form $$y = a x ^ { \frac { 7 } { 2 } } + b x ^ { \frac { 5 } { 2 } } + c x ^ { \frac { 3 } { 2 } } + d x ^ { \frac { 1 } { 2 } }$$ where \(a , b , c\) and \(d\) are fully simplified constants. The curve \(C\) has three turning points.
   Using calculus,
2. show that the \(x\) coordinates of the three turning points satisfy the equation $$2 x ^ { 3 } - 10 x ^ { 2 } + 13 x - 5 = 0$$ Given that the \(x\) coordinate of one of the turning points is 1
3. find, using algebra, the exact \(x\) coordinates of the other two turning points.
   (Solutions based entirely on calculator technology are not acceptable.)

6 A curve has equation \(y = \mathrm { e } ^ { x ^ { 2 } + 3 x }\).

1. Determine the \(x\)-coordinates of any stationary points on the curve.
2. Show that the curve is convex for all values of \(x\).

3. The curve \(C\) has equation $$y = 8 \sqrt { x } + \frac { 18 } { \sqrt { x } } - 20 \quad x > 0$$ a. Find
i) \(\frac { d y } { d x }\) ii) \(\frac { d ^ { 2 } y } { d x ^ { 2 } }\) b. Use calculus to find the coordinates of the stationary point of \(C\).
c. Determine whether the stationary point is a maximum or minimum, giving a reason for your answer.

1. A curve \(C\) has equation

$$y = x ^ { 2 } - 2 x - 24 \sqrt { x } , \quad x > 0$$

1. Find (i) \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) (ii) \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)
2. Verify that \(C\) has a stationary point when \(x = 4\)
3. Determine the nature of this stationary point, giving a reason for your answer.

7 At the point \(( x , y )\), where \(x > 0\), the gradient of a curve is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { \frac { 1 } { 2 } } + \frac { 16 } { x ^ { 2 } } - 7$$

1. 1. Verify that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) when \(x = 4\).
      (1 mark)
   2. Write \(\frac { 16 } { x ^ { 2 } }\) in the form \(16 x ^ { k }\), where \(k\) is an integer.
   3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
   4. Hence determine whether the point where \(x = 4\) is a maximum or a minimum, giving a reason for your answer.
2. The point \(P ( 1,8 )\) lies on the curve.
   1. Show that the gradient of the curve at the point \(P\) is 12 .
   2. Find an equation of the normal to the curve at \(P\).
3. 1. Find \(\int \left( 3 x ^ { \frac { 1 } { 2 } } + \frac { 16 } { x ^ { 2 } } - 7 \right) \mathrm { d } x\).
   2. Hence find the equation of the curve which passes through the point \(P ( 1,8 )\).

2 The surface \(S\) has equation \(z = x ^ { 2 } y - 8 x y ^ { 2 } + \frac { x } { y }\) for \(y \neq 0\).

1. (a) Find the following.

6 A curve \(C\) is defined for \(x > 0\) by the equation \(y = x + 1 + \frac { 4 } { x ^ { 2 } }\) and is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{c16d94a6-52f2-4bf3-acee-0b227ae55a1a-4_545_784_420_628}

1. 1. Given that \(y = x + 1 + \frac { 4 } { x ^ { 2 } }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   2. The curve \(C\) has a minimum point \(M\). Find the coordinates of \(M\).
   3. Find an equation of the normal to \(C\) at the point ( 1,6 ).
2. 1. Find \(\int \left( x + 1 + \frac { 4 } { x ^ { 2 } } \right) \mathrm { d } x\).
   2. Hence find the area of the region bounded by the curve \(C\), the lines \(x = 1\) and \(x = 4\) and the \(x\)-axis.