1.07n Stationary points: find maxima, minima using derivatives

8. \includegraphics[max width=\textwidth, alt={}, center]{a86f277c-a2ec-4ba0-ab08-575cad2a5e53-3_424_698_246_479} The diagram shows the curve \(y = \mathrm { f } ( x )\) in the interval \(0 \leq x \leq 2 \pi\) where $$\mathrm { f } ( x ) = \frac { \cos x } { 2 - \sin x } , \quad x \in \mathbb { R }$$

1. Show that \(\mathrm { f } ^ { \prime } ( x ) = \frac { 1 - 2 \sin x } { ( 2 - \sin x ) ^ { 2 } }\).
2. Find an equation for the tangent to the curve \(y = \mathrm { f } ( x )\) at the point where \(x = \pi\).
3. Find the minimum and maximum values of \(\mathrm { f } ( x )\) in the interval \(0 \leq x \leq 2 \pi\).
4. Explain why your answers to part (c) are the minimum and maximum values of \(\mathrm { f } ( x )\) for all real values of \(x\).

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

1. State the equation of the asymptote of the curve.
2. Show that \(- \frac { 1 } { 6 } \leqslant y \leqslant \frac { 1 } { 2 }\).

7 The curve with equation $$y = \frac { x } { \cosh x }$$ has one stationary point for \(x > 0\).

1. Show that the \(x\)-coordinate of this stationary point satisfies the equation \(x \tanh x - 1 = 0\). The positive root of the equation \(x \tanh x - 1 = 0\) is denoted by \(\alpha\).
2. Draw a sketch showing (for positive values of \(x\) ) the graph of \(y = \tanh x\) and its asymptote, and the graph of \(y = \frac { 1 } { x }\). Explain how you can deduce from your sketch that \(\alpha > 1\).
3. Use the Newton-Raphson method, taking first approximation \(x _ { 1 } = 1\), to find further approximations \(x _ { 2 }\) and \(x _ { 3 }\) for \(\alpha\).
4. By considering the approximate errors in \(x _ { 1 }\) and \(x _ { 2 }\), estimate the error in \(x _ { 3 }\).

5 A curve has equation \(y = \frac { x ^ { 3 } - k ^ { 3 } } { x ^ { 2 } - 4 }\), where \(k\) is a positive constant and \(k \neq 2\).

1. Find the equations of the three asymptotes.
2. Use your graphical calculator to obtain rough sketches of the curve in the two separate cases \(k < 2\) and \(k > 2\).
3. In the case \(k < 2\), your sketch may not show clearly the shape of the curve near \(x = 0\). Use calculus to show that the curve has a minimum point when \(x = 0\).
4. In the case \(k > 2\), your sketch may not show clearly how the curve approaches its asymptote as \(x \rightarrow + \infty\). Show algebraically that the curve crosses this asymptote.
5. Use the results of parts (iii) and (iv) to produce more accurate sketches of the curve in the two separate cases \(k < 2\) and \(k > 2\). These sketches should indicate where the curve crosses the axes, and should show clearly how the curve approaches its asymptotes. The presence of stationary points should be clearly shown, but there is no need to find their coordinates. RECOGNISING ACHIEVEMENT \section*{OXFORD CAMBRIDGE AND RSA EXAMINATIONS} \section*{MEI STRUCTURED MATHEMATICS} Further Methods for Advanced Mathematics (FP2)
   Tuesday

5 A curve has equation \(y = \frac { x ^ { 3 } - k ^ { 3 } } { x ^ { 2 } - 4 }\), where \(k\) is a positive constant and \(k \neq 2\).

1. Find the equations of the three asymptotes.
2. Use your graphical calculator to obtain rough sketches of the curve in the two separate cases \(k < 2\) and \(k > 2\).
3. In the case \(k < 2\), your sketch may not show clearly the shape of the curve near \(x = 0\). Use calculus to show that the curve has a minimum point when \(x = 0\).
4. In the case \(k > 2\), your sketch may not show clearly how the curve approaches its asymptote as \(x \rightarrow + \infty\). Show algebraically that the curve crosses this asymptote.
5. Use the results of parts (iii) and (iv) to produce more accurate sketches of the curve in the two separate cases \(k < 2\) and \(k > 2\). These sketches should indicate where the curve crosses the axes, and should show clearly how the curve approaches its asymptotes. The presence of stationary points should be clearly shown, but there is no need to find their coordinates.

5 A curve has parametric equations \(x = \frac { t ^ { 2 } } { 1 + t ^ { 2 } } , y = t ^ { 3 } - \lambda t\), where \(\lambda\) is a constant.

1. Use your calculator to obtain a sketch of the curve in each of the cases $$\lambda = - 1 , \quad \lambda = 0 \quad \text { and } \quad \lambda = 1 .$$ Name any special features of these curves.
2. By considering the value of \(x\) when \(t\) is large, write down the equation of the asymptote. For the remainder of this question, assume that \(\lambda\) is positive.
3. Find, in terms of \(\lambda\), the coordinates of the point where the curve intersects itself.
4. Show that the two points on the curve where the tangent is parallel to the \(x\)-axis have coordinates $$\left( \frac { \lambda } { 3 + \lambda } , \pm \sqrt { \frac { 4 \lambda ^ { 3 } } { 27 } } \right)$$ Fig. 5 shows a curve which intersects itself at the point ( 2,0 ) and has asymptote \(x = 8\). The stationary points A and B have \(y\)-coordinates 2 and - 2 . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{43b4c7ed-3556-4d87-8aef-0111fe747982-4_791_609_1482_769} \captionsetup{labelformat=empty} \caption{Fig. 5}
   \end{figure}
5. For the curve sketched in Fig. 5, find parametric equations of the form \(x = \frac { a t ^ { 2 } } { 1 + t ^ { 2 } } , y = b \left( t ^ { 3 } - \lambda t \right)\), where \(a , \lambda\) and \(b\) are to be determined.

3.Given that \(\phi = \frac { 1 } { 2 } ( \sqrt { 5 } + 1 )\) ,

1. show that
   1. \(\phi ^ { 2 } = \phi + 1\)
   2. \(\frac { 1 } { \phi } = \phi - 1\)
2. The equations of two curves are $$\begin{array} { r l r l } y & = \frac { 1 } { x } & x > 0 \\ \text { and } & y & = \ln x - x + k & x > 0 \end{array}$$ where \(k\) is a positive constant.
   The curves touch at the point \(P\) .
   Find in terms of \(\phi\)
   1. the coordinates of \(P\) ,
   2. the value of \(k\) .

7.Figure 2 shows a rectangular section of marshland,\(O A B C\) ,which is \(a\) metres long by \(b\) metres wide,where \(a > b\) . \begin{figure}[h] \end{figure} Edgar intends to get from \(O\) to \(B\) in the shortest possible time.In order to do this,he runs along edge \(O A\) for a distance \(x\) metres \(( 0 \leqslant x < a )\) to the point \(D\) before wading through the marsh directly from \(D\) to \(B\) . Edgar can wade through the marsh at a constant speed of \(1 \mathrm {~ms} ^ { - 1 }\) ,and he can run along the edge of the marsh at a constant speed of \(\lambda \mathrm { ms } ^ { - 1 }\) ,where \(\lambda > 1\)

1. By finding an expression in terms of \(x\) for the time taken,\(t\) seconds,for Edgar to reach \(B\) from \(O\) ,show that $$\frac { \mathrm { d } t } { \mathrm {~d} x } = \frac { 1 } { \lambda } - \frac { a - x } { \sqrt { ( a - x ) ^ { 2 } + b ^ { 2 } } }$$
2. 1. Find,in terms of \(a , b\) and \(\lambda\) ,the value of \(x\) for which \(\frac { \mathrm { d } t } { \mathrm {~d} x } = 0\)
   2. Show that this value of \(x\) lies in the interval \(0 \leqslant x < a\) provided \(\lambda \geqslant \sqrt { 1 + \frac { b ^ { 2 } } { a ^ { 2 } } }\)
   3. For \(\lambda\) in this range,show that the value of \(x\) found in(b)(i)gives a minimum value of \(t\) .
3. Find the minimum time taken for Edgar to get from \(O\) to \(B\) if
   1. \(\lambda \geqslant \sqrt { 1 + \frac { b ^ { 2 } } { a ^ { 2 } } }\)
   2. \(1 < \lambda < \sqrt { 1 + \frac { b ^ { 2 } } { a ^ { 2 } } }\) Edgar's friend,Frankie,also runs at a constant speed of \(\lambda \mathrm { m } \mathrm { s } ^ { - 1 }\) .Frankie runs along the edges \(O A\) and \(A B\) .Given that \(\lambda \geqslant \sqrt { 1 + \frac { b ^ { 2 } } { a ^ { 2 } } }\)
4. find the range of values of \(\lambda\) for which Frankie gets to \(B\) from \(O\) in a shorter time than Edgar's minimum time.

1. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = 1 + \frac { 4 } { x ( x - 3 ) }$$ The curve has a turning point at the point \(P\), and the lines with equations \(y = 1 , x = 0\) and \(x = a\) are asymptotes to the curve.

1. Write down the value of \(a\).
2. Find the coordinates of \(P\), justifying your answer.
3. Sketch the curve with equation \(y = \left| \mathrm { f } \left( x + \frac { 3 } { 2 } \right) \right| - 1\) On your sketch, you should show the coordinates of any points of intersection with the coordinate axes, the coordinates of any turning points and the equations of any asymptotes. \includegraphics[max width=\textwidth, alt={}, center]{4d5b914c-28b2-4485-a42e-627c95fa16e2-02_2255_50_311_1980}

4. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the parabola with equation \(y = \frac { 1 } { 2 } x ( 10 - x ) , 0 \leqslant x \leqslant 10\) This question concerns rectangles that lie under the parabola in the first quadrant.The bottom edge of each rectangle lies along the \(x\)-axis and the top left vertex lies on the parabola.Some examples are shown in Figure 2. Let the \(x\) coordinate of the top left vertex be \(a\) .

1. Explain why the width,\(w\) ,of such a rectangle must satisfy \(w \leqslant 10 - 2 a\)
2. Find the value of \(a\) that gives the maximum area for such a rectangle. Given that the rectangle must be a square,
3. find the value of \(a\) that gives the maximum area for such a square. Given that the area of the rectangles is fixed as 36
4. find the range of possible values for \(a\) . \includegraphics[max width=\textwidth, alt={}, center]{4d5b914c-28b2-4485-a42e-627c95fa16e2-16_2255_50_311_1980}

1. $$\mathrm { f } ( x ) = x ^ { \left( x ^ { 2 } \right) } \quad x > 0$$ Use logarithms to find the \(x\) coordinate of the stationary point of the curve with equation \(y = \mathrm { f } ( x )\) .

4.A rectangle \(A B C D\) is drawn so that \(A\) and \(B\) lie on the \(x\)-axis,and \(C\) and \(D\) lie on the curve with equation \(y = \cos x , - \frac { \pi } { 2 } < x < \frac { \pi } { 2 }\) .The point \(A\) has coordinates \(( p , 0 )\) ,where \(0 < p < \frac { \pi } { 2 }\) .

1. Find an expression,in terms of \(p\) ,for the area of this rectangle. The maximum area of \(A B C D\) is \(S\) and occurs when \(p = \alpha\) .Show that
2. \(\frac { \pi } { 4 } < \alpha < 1\) ,
3. \(S = \frac { 2 \alpha ^ { 2 } } { \sqrt { } \left( 1 + \alpha ^ { 2 } \right) }\) ,
4. \(\frac { \pi ^ { 2 } } { 2 \sqrt { } \left( 16 + \pi ^ { 2 } \right) } < S < \sqrt { } 2\) .

7. \includegraphics[max width=\textwidth, alt={}, center]{0df09d8a-7478-4679-b117-128ee226db6a-5_648_1590_296_275} The circle \(C _ { 1 }\) has centre \(O\) and radius \(R\). The tangents \(A P\) and \(B P\) to \(C _ { 1 }\) meet at the point \(P\) and angle \(A P B = 2 \alpha , 0 < \alpha < \frac { \pi } { 2 }\). A sequence of circles \(C _ { 1 } , C _ { 2 } , \ldots , C _ { n } , \ldots\) is drawn so that each new circle \(C _ { n + 1 }\) touches each of \(C _ { n } , A P\) and \(B P\) for \(n = 1,2,3 , \ldots\) as shown in Figure 2. The centre of each circle lies on the line \(O P\).

1. Show that the radii of the circles form a geometric sequence with common ratio $$\frac { 1 - \sin \alpha } { 1 + \sin \alpha }$$
2. Find, in terms of \(R\) and \(\alpha\), the total area enclosed by all the circles, simplifying your answer. The area inside the quadrilateral \(P A O B\), not enclosed by part of \(C _ { 1 }\) or any of the other circles, is \(S\).
3. Show that $$S = R ^ { 2 } \left( \alpha + \cot \alpha - \frac { \pi } { 4 } \operatorname { cosec } \alpha - \frac { \pi } { 4 } \sin \alpha \right) .$$
4. Show that, as \(\alpha\) varies, $$\frac { \mathrm { d } S } { \mathrm {~d} \alpha } = R ^ { 2 } \cot ^ { 2 } \alpha \left( \frac { \pi } { 4 } \cos \alpha - 1 \right)$$
5. Find, in terms of \(R\), the least value of \(S\) for \(\frac { \pi } { 6 } \leq \alpha \leq \frac { \pi } { 4 }\).

1. (a) The function \(\mathrm { f } ( x )\) has \(\mathrm { f } ^ { \prime } ( x ) = \frac { \mathrm { u } ( x ) } { \mathrm { v } ( x ) }\). Given that \(\mathrm { f } ^ { \prime } ( k ) = 0\), show that \(\mathrm { f } ^ { \prime \prime } ( k ) = \frac { \mathrm { u } ^ { \prime } ( k ) } { \mathrm { v } ( k ) }\).

\begin{figure}[h] \end{figure} (b) The curve \(C\) with equation $$y = \frac { 2 x ^ { 2 } + 3 } { x ^ { 2 } - 1 }$$ crosses the \(y\)-axis at the point \(A\). Figure 1 shows a sketch of \(C\) together with its 3 asymptotes.

1. Find the coordinates of the point \(A\).
2. Find the equations of the asymptotes of \(C\). The point \(P ( a , b ) , a > 0\) and \(b > 0\), lies on \(C\). The point \(Q\) also lies on \(C\) with \(P Q\) parallel to the \(x\)-axis and \(A P = A Q\).
3. Show that the area of triangle \(P A Q\) is given by \(\frac { 5 a ^ { 3 } } { a ^ { 2 } - 1 }\).
4. Find, as \(a\) varies, the minimum area of triangle \(P A Q\), giving your answer in its simplest form.

6.(a)Given that \(x ^ { 4 } + y ^ { 4 } = 1\) ,prove that \(x ^ { 2 } + y ^ { 2 }\) is a maximum when \(x = \pm y\) ,and find the maximum and minimum values of \(x ^ { 2 } + y ^ { 2 }\) .
(b)On the same diagram,sketch the curves \(C _ { 1 }\) and \(C _ { 2 }\) with equations \(x ^ { 4 } + y ^ { 4 } = 1\) and \(x ^ { 2 } + y ^ { 2 } = 1\) respectively.
(c)Write down the equation of the circle \(C _ { 3 }\) ,centre the origin,which touches the curve \(C _ { 1 }\) at the points where \(x = \pm y\) .

7. $$\mathrm { f } ( x ) = \left[ 1 + \cos \left( x + \frac { \pi } { 4 } \right) \right] \left[ 1 + \sin \left( x + \frac { \pi } { 4 } \right) \right] , \quad 0 \leqslant x \leqslant 2 \pi$$

1. Show that \(\mathrm { f } ( x )\) may be written in the form $$f ( x ) = \left( \frac { 1 } { \sqrt { 2 } } + \cos x \right) ^ { 2 } , \quad 0 \leqslant x \leqslant 2 \pi$$
2. Find the range of the function \(\mathrm { f } ( x )\). The graph of \(y = \mathrm { f } ( x )\) is shown in Figure 2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{0396f61a-b844-40ed-98d1-82ee2d8a6807-5_426_938_849_591} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure}
3. Find the coordinates of all the maximum and minimum points on this curve. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{0396f61a-b844-40ed-98d1-82ee2d8a6807-5_432_942_1535_589} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} The region \(R\), bounded by \(y = 2\) and \(y = \mathrm { f } ( x )\), is shown shaded in Figure 3.
4. Find the area of \(R\).

7. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C _ { 1 }\) with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \frac { x } { 3 } + \frac { 12 } { x } \quad x \neq 0$$ The lines \(x = 0\) and \(y = \frac { x } { 3 }\) are asymptotes to \(C _ { 1 }\). The point \(A\) on \(C _ { 1 }\) is a minimum and the point \(B\) on \(C _ { 1 }\) is a maximum.

1. Find the coordinates of \(A\) and \(B\). There is a normal to \(C _ { 1 }\), which does not intersect \(C _ { 1 }\) a second time, that has equation \(x = k\), where \(k > 0\).
2. Write down the value of \(k\). The point \(P ( \alpha , \beta ) , \alpha > 0\) and \(\alpha \neq k\), lies on \(C _ { 1 }\). The normal to \(C _ { 1 }\) at \(P\) does not intersect \(C _ { 1 }\) a second time.
3. Find the value of \(\alpha\), leaving your answer in simplified surd form.
4. Find the equation of this normal. The curve \(C _ { 2 }\) has equation \(y = | \mathrm { f } ( x ) |\)
5. Sketch \(C _ { 2 }\) stating the coordinates of any turning points and the equations of any asymptotes. The line with equation \(y = m x + 1\) does not touch or intersect \(C _ { 2 }\).
6. Find the set of possible values for \(m\).

4. \begin{figure}[h] \end{figure} Figure 1 shows the equilateral triangle \(L M N\) of side 2 cm .The point \(P\) lies on \(L M\) such that \(L P = x \mathrm {~cm}\) and the point \(Q\) lies on \(L N\) such that \(L Q = y \mathrm {~cm}\) .The points \(P\) and \(Q\) are chosen so that the area of triangle \(L P Q\) is half the area of triangle \(L M N\) .

1. Show that \(x y = 2\)
2. Find the shortest possible length of \(P Q\) ,justifying your answer. Mathematicians know that for any closed curve or polygon enclosing a fixed area,the ratio \(\frac { \text { area enclosed } } { \text { perimeter } }\) is a maximum when the closed curve is a circle. By considering 6 copies of triangle \(L M N\) suitably arranged,
3. find the length of the shortest line or curve that can be drawn from a point on \(L M\) to a point on \(L N\) to divide the area of triangle \(L M N\) in half.Justify your answer.
   (6)

5 In this question, you are required to investigate the curve with equation $$y = x ^ { m } ( 1 - x ) ^ { n } , \quad 0 \leqslant x \leqslant 1 ,$$ for various positive values of \(m\) and \(n\).

1. On separate diagrams, sketch the curve in each of the following cases.
   (A) \(m = 1 , n = 1\),
   (B) \(m = 2 , n = 2\),
   (C) \(m = 2 , n = 4\),
   (D) \(m = 4 , n = 2\).
2. What feature does the curve have when \(m = n\) ? What is the effect on the curve of interchanging \(m\) and \(n\) when \(m \neq n\) ?
3. Describe how the \(x\)-coordinate of the maximum on the curve varies as \(m\) and \(n\) vary. Use calculus to determine the \(x\)-coordinate of the maximum.
4. Find the condition on \(m\) for the gradient to be zero when \(x = 0\). State a corresponding result for the gradient to be zero when \(x = 1\).
5. Use your calculator to investigate the shape of the curve for large values of \(m\) and \(n\). Hence conjecture what happens to the value of the integral \(\int _ { 0 } ^ { 1 } x ^ { m } ( 1 - x ) ^ { n } \mathrm {~d} x\) as \(m\) and \(n\) tend to infinity.
6. Use your calculator to investigate the shape of the curve for small values of \(m\) and \(n\). Hence conjecture what happens to the shape of the curve as \(m\) and \(n\) tend to zero. }{www.ocr.org.uk}) after the live examination series.
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9 The curve \(y = x ^ { 3 } + p x ^ { 2 } + 2\) has a stationary point when \(x = 4\). Find the value of the constant \(p\) and determine whether the stationary point is a maximum or minimum point.

8

1. Find the equation of the tangent to the curve \(y = 7 + 6 x - x ^ { 2 }\) at the point \(P\) where \(x = 5\), giving your answer in the form \(a x + b y + c = 0\).
2. This tangent meets the \(x\)-axis at \(Q\). Find the coordinates of the mid-point of \(P Q\).
3. Find the equation of the line of symmetry of the curve \(y = 7 + 6 x - x ^ { 2 }\).
4. State the set of values of \(x\) for which \(7 + 6 x - x ^ { 2 }\) is an increasing function.

7 A curve has equation \(y = ( x + 2 ) \left( x ^ { 2 } - 3 x + 5 \right)\).

1. Find the coordinates of the minimum point, justifying that it is a minimum.
2. Calculate the discriminant of \(x ^ { 2 } - 3 x + 5\).
3. Explain why \(( x + 2 ) \left( x ^ { 2 } - 3 x + 5 \right)\) is always positive for \(x > - 2\).

10

1. Solve the equation \(9 x ^ { 2 } + 18 x - 7 = 0\).
2. Find the coordinates of the stationary point on the curve \(y = 9 x ^ { 2 } + 18 x - 7\).
3. Sketch the curve \(y = 9 x ^ { 2 } + 18 x - 7\), giving the coordinates of all intercepts with the axes.
4. For what values of \(x\) does \(9 x ^ { 2 } + 18 x - 7\) increase as \(x\) increases?

8

1. Express \(2 x ^ { 2 } + 5 x\) in the form \(2 ( x + p ) ^ { 2 } + q\).
2. State the coordinates of the minimum point of the curve \(y = 2 x ^ { 2 } + 5 x\).
3. State the equation of the normal to the curve at its minimum point.
4. Solve the inequality \(2 x ^ { 2 } + 5 x > 0\).

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.