1.07n Stationary points: find maxima, minima using derivatives

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

Solutions relying entirely on calculator technology are not acceptable.
The curve \(C\) has equation $$y = 4 x ^ { \frac { 1 } { 2 } } + 9 x ^ { - \frac { 1 } { 2 } } + 3 \quad x > 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) giving each term in simplest form.
2. Hence find the \(x\) coordinate of the stationary point of \(C\).
3. 1. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) giving each term in simplest form.
   2. Hence determine the nature of the stationary point of \(C\), giving a reason for your answer.
4. State the range of values of \(x\) for which \(y\) is decreasing.

7. \begin{figure}[h] \end{figure} The curve \(C\) has equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = x ^ { 3 } \sqrt { 4 x + 7 } \quad x \geqslant - \frac { 7 } { 4 }$$

1. Show that $$\mathrm { f } ^ { \prime } ( x ) = \frac { k x ^ { 2 } ( 2 x + 3 ) } { \sqrt { 4 x + 7 } }$$ where \(k\) is a constant to be found. The point \(P\), shown in Figure 3, is the minimum turning point on \(C\).
2. Find the coordinates of \(P\).
3. Hence find the range of the function g defined by $$g ( x ) = - 4 f ( x ) \quad x \geqslant - \frac { 7 } { 4 }$$ The point \(Q\) with coordinates \(\left( \frac { 1 } { 2 } , \frac { 3 } { 8 } \right)\) lies on \(C\).
4. Find the coordinates of the point to which \(Q\) is mapped when \(C\) is transformed to the curve with equation $$y = 40 \mathrm { f } \left( x - \frac { 3 } { 2 } \right) - 8$$

4 The curve \(C\) has equation \(y = \frac { x + 1 } { x ^ { 2 } + 3 }\).

1. By considering a suitable quadratic equation in \(x\), find the set of possible values of \(y\) for points on \(C\).
2. Deduce the coordinates of the turning points on \(C\).

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

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Hence find the coordinates of the stationary points on the curve.

8 A curve has equation \(y = 2 x ^ { 3 } - 5 x ^ { 2 } - 4 x + 1\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Hence find the \(x\)-coordinates of the stationary points of the curve.
3. By using the second derivative, determine whether each of the stationary points is a maximum or a minimum.

10 A curve has equation \(y = \frac { \mathrm { e } ^ { x } } { x ^ { 2 } }\). Show that

1. the gradient of the curve at \(x = 1\) is - e ,
2. there is a stationary point at \(x = 2\) and determine its nature.

12 A curve \(C\) is given by the parametric equations \(x = 2 \tan \theta , y = 1 + \operatorname { cosec } \theta\) for \(0 < \theta < 2 \pi , \theta \neq \frac { 1 } { 2 } \pi , \pi , \frac { 3 } { 2 } \pi\).

1. Show that the cartesian equation for \(C\) is \(\frac { 4 } { x ^ { 2 } } = y ^ { 2 } - 2 y\).
2. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence show that \(C\) has no stationary points.
3. \(P\) is the point on \(C\) where \(\theta = \frac { 1 } { 4 } \pi\). The tangent to \(C\) at \(P\) intersects the \(y\)-axis at \(Q\) and the \(x\)-axis at \(R\). Find the exact area of triangle \(O Q R\).

8 Show that the graph of \(y = x ^ { 2 } - \ln x\) has only one stationary point and give the coordinates of that point in exact form.

5 The curve \(C\) has equation \(y = \frac { 12 ( x + 1 ) } { ( x - 2 ) ^ { 2 } }\).

1. Determine the coordinates of any stationary points of \(C\).
2. Sketch \(C\).

3 A curve has equation \(y = \frac { 2 x ^ { 2 } - x - 1 } { 2 x - 3 }\).

1. Show that the curve meets the line \(y = k\) when \(2 x ^ { 2 } - ( 2 k + 1 ) x + ( 3 k - 1 ) = 0\), and hence show that no part of the curve exists in the interval \(\frac { 1 } { 2 } < y < \frac { 9 } { 2 }\).
2. Deduce the coordinates of the turning points of this curve.

6

1. Find the coordinates of the stationary points of the curve with equation $$y = 3 x ^ { 4 } - 20 x ^ { 3 } + 36 x ^ { 2 }$$ and determine their nature.
2. Sketch the graph of \(y = 3 x ^ { 4 } - 20 x ^ { 3 } + 36 x ^ { 2 }\) and hence state the set of values of \(k\) for which the equation \(3 x ^ { 4 } - 20 x ^ { 3 } + 36 x ^ { 2 } = k\) has exactly four distinct real roots.

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

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Hence find the coordinates of the stationary points on the curve.

10 A curve has equation $$y = \mathrm { e } ^ { a x } \cos b x$$ where \(a\) and \(b\) are constants.

1. Show that, at any stationary points on the curve, \(\tan b x = \frac { a } { b }\).
2. \includegraphics[max width=\textwidth, alt={}, center]{ac5bf967-8b97-4bf3-991f-28c3ec7a25da-4_622_896_957_333} Values of related quantities \(x\) and \(y\) were measured in an experiment and plotted on a graph of \(y\) against \(x\), as shown in the diagram. Two of the points, labelled \(A\) and \(B\), have coordinates \(( 0,1 )\) and \(( 0.2 , - 0.8 )\) respectively. A third point labelled C has coordinates ( \(0.3,0.04\) ). Attempts were then made to find the equation of a curve which fitted closely to these three points, and two models were proposed. In the first model the equation is \(y = \mathrm { e } ^ { - x } \cos 15 x\).
   In the second model the equation is \(y = f \cos ( \lambda x ) + \mathrm { g }\), where the constants \(f , \lambda\), and \(g\) are chosen to give a maximum precisely at the point \(A ( 0,1 )\) and a minimum precisely at the point \(B ( 0.2 , - 0.8 )\). By calculating suitable values evaluate the suitability of the two models.

5 \includegraphics[max width=\textwidth, alt={}, center]{1c957cfe-bead-41d9-8985-479e876e1616-3_577_743_287_662} The diagram shows a sector of a circle, \(O M N\). The angle \(M O N\) is \(2 x\) radians, the radius of the circle is \(r\) and \(O\) is the centre.

1. Find expressions, in terms of \(r\) and \(x\), for the area, \(A\), and the perimeter, \(P\), of the sector.
2. Given that \(P = 20\), show that \(A = \frac { 100 x } { ( 1 + x ) ^ { 2 } }\).
3. Find \(\frac { \mathrm { d } A } { \mathrm {~d} x }\), and hence find the value of \(x\) for which the area of the sector is a maximum.

10 A curve has equation $$y = \mathrm { e } ^ { a x } \cos b x$$ where \(a\) and \(b\) are constants.

1. Show that, at any stationary points on the curve, \(\tan b x = \frac { a } { b }\).
2. \includegraphics[max width=\textwidth, alt={}, center]{1c957cfe-bead-41d9-8985-479e876e1616-4_620_896_959_333} Values of related quantities \(x\) and \(y\) were measured in an experiment and plotted on a graph of \(y\) against \(x\), as shown in the diagram. Two of the points, labelled \(A\) and \(B\), have coordinates \(( 0,1 )\) and \(( 0.2 , - 0.8 )\) respectively. A third point labelled C has coordinates ( \(0.3,0.04\) ). Attempts were then made to find the equation of a curve which fitted closely to these three points, and two models were proposed. In the first model the equation is \(y = \mathrm { e } ^ { - x } \cos 15 x\).
   In the second model the equation is \(y = f \cos ( \lambda x ) + \mathrm { g }\), where the constants \(f , \lambda\), and \(g\) are chosen to give a maximum precisely at the point \(A ( 0,1 )\) and a minimum precisely at the point \(B ( 0.2 , - 0.8 )\). By calculating suitable values evaluate the suitability of the two models.

6 The curve \(C\) has equation \(y = \frac { x + 1 } { x ^ { 2 } + 3 }\).

1. By considering a suitable quadratic equation in \(x\), find the set of possible values of \(y\) for points on \(C\).
2. Deduce the coordinates of the turning points on \(C\).
3. Sketch \(C\).

6 The curve \(S\) has equation \(y = \frac { x ^ { 2 } + 1 } { ( x + 1 ) ^ { 2 } }\).

1. Write down the equations of the asymptotes of \(S\).
2. Determine \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the coordinates of any turning points of \(S\).
3. Sketch \(S\).

9

1. Show that \(\int x ( x - 2 ) ^ { \frac { 3 } { 2 } } \mathrm {~d} x = \frac { 2 } { 35 } ( 5 x + 4 ) ( x - 2 ) ^ { \frac { 5 } { 2 } } + c\).
2. Hence find the coordinates of the stationary points of the curve $$y = \frac { 2 } { 35 } ( 5 x + 4 ) ( x - 2 ) ^ { \frac { 5 } { 2 } } + x ^ { 2 } - \frac { 1 } { 3 } x ^ { 3 }$$

7 Find the coordinates of the two stationary points of the curve $$9 x ^ { 2 } + 4 y ^ { 2 } - 6 x - 4 y = 34$$ showing that one is a maximum and one is a minimum.

9

1. Find the coordinates of the stationary point of the curve with equation $$y = \ln x - k x , \text { where } k > 0 \text { and } x > 0$$ and determine its nature.
2. Hence show that the equation \(\ln x - k x = 0\) has real roots if \(0 < k \leqslant \frac { 1 } { \mathrm { e } }\).
3. In the particular case that \(k = \frac { 1 } { 3 }\), the equation \(\ln x - k x = 0\) has two roots, one of which is near \(x = 5\). Use the Newton-Raphson process to find, correct to 3 significant figures, the root of the equation \(\ln x - \frac { 1 } { 3 } x = 0\) which is near \(x = 5\).
4. Show that the equation \(\ln x - k x = 0\) has one real root if \(k \leqslant 0\).
5. Explain why the equation \(\ln x - k x = 0\) has two distinct real roots if \(0 < k < \frac { 1 } { \mathrm { e } }\).

10 A particle \(P\) moves in a straight line starting from \(O\). At time \(t\) seconds after leaving \(O\), the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 5 + 1.5 t - 0.125 t ^ { 3 }\).

1. Find the displacement of \(P\) between the times \(t = 1\) and \(t = 4\).
2. Find the time at which the velocity of \(P\) is a maximum, justifying your answer.

6 The curve \(C\) has equation \(y = \frac { x + 1 } { x ^ { 2 } + 3 }\).

1. By considering a suitable quadratic equation in \(x\), find the set of possible values of \(y\) for points on \(C\).
2. Deduce the coordinates of the turning points on \(C\).
3. Sketch \(C\).

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

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Hence find the coordinates of the stationary points on the curve.

5 \includegraphics[max width=\textwidth, alt={}, center]{8a0a6e46-99cf-4217-93ad-5ed6e9d7c4ef-3_565_730_219_669} The diagram shows a sector of a circle, \(O M N\). The angle \(M O N\) is \(2 x\) radians, the radius of the circle is \(r\) and \(O\) is the centre.

1. Find expressions, in terms of \(r\) and \(x\), for the area, \(A\), and the perimeter, \(P\), of the sector.
2. Given that \(P = 20\), show that \(A = \left( \frac { 10 } { 1 + x } \right) ^ { 2 }\).
3. Find \(\frac { \mathrm { d } A } { \mathrm {~d} x }\), and hence find the value of \(x\) for which the area of the sector is a maximum.

7 A cubic polynomial is given by $$\mathrm { P } ( x ) = x ^ { 3 } - 3 x ^ { 2 } + a x + b$$ where \(a\) and \(b\) are constants.

1. If \(\mathrm { P } ( x )\) is exactly divisible by \(x - 1\), and has a local maximum at \(x = - 1\), determine the values of \(a\) and \(b\).
2. Sketch the curve \(y = \mathrm { P } ( x )\), marking the intercepts and the \(x\)-coordinates of the stationary points.
3. Expand and simplify \(\mathrm { P } ( 1 + x )\), and deduce that \(\mathrm { P } ( 1 + x ) = - \mathrm { P } ( 1 - x )\). Interpret this result graphically.