1.07n Stationary points: find maxima, minima using derivatives

Find the coordinates of the stationary points of the curve with equation $$x^3 + y^3 - 3xy = 48$$ and determine their nature. [14]

A student was attempting to prove that \(x = \frac{1}{2}\) is the only real root of $$x^3 + \frac{1}{4}x - \frac{1}{2} = 0.$$ The attempted solution was as follows. $$x^3 + \frac{1}{4}x = \frac{1}{2}$$ $$\therefore \quad x(x^2 + \frac{1}{4}) = \frac{1}{2}$$ $$\therefore \quad x = \frac{1}{2}$$ or $$x^2 + \frac{1}{4} = \frac{1}{2}$$ i.e. $$x^2 = -\frac{1}{4} \quad \text{no solution}$$ $$\therefore \quad \text{only real root is } x = \frac{1}{2}$$

1. Explain clearly the error in the above attempt. [2]
2. Give a correct proof that \(x = \frac{1}{2}\) is the only real root of \(x^3 + \frac{1}{4}x - \frac{1}{2} = 0\). [3]

The equation $$x^3 + \beta x - \alpha = 0 \quad \text{(I)}$$ where \(\alpha\), \(\beta\) are real, \(\alpha \neq 0\), has a real root at \(x = \alpha\).

1. Find and simplify an expression for \(\beta\) in terms of \(\alpha\) and prove that \(\alpha\) is the only real root provided \(|\alpha| < 2\). [6]

An examiner chooses a positive number \(\alpha\) so that \(\alpha\) is the only real root of equation (I) but the incorrect method used by the student produces 3 distinct real "roots".

1. Find the range of possible values for \(\alpha\). [7]

$$f(x) = x^3 - (k+4)x + 2k,$$ where \(k\) is a constant.

1. Show that, for all values of \(k\), the curve with equation \(y = f(x)\) passes through the point \((2, 0)\). [1]
2. Find the values of \(k\) for which the equation \(f(x) = 0\) has exactly two distinct roots. [5]

Given that \(k > 0\), that the \(x\)-axis is a tangent to the curve with equation \(y = f(x)\), and that the line \(y = p\) intersects the curve in three distinct points,

1. find the set of values that \(p\) can take. [5]

\includegraphics{figure_1} Figure 1 shows a sketch of the curve \(C\) with equation $$y = \cos x \ln(\sec x), \quad -\frac{\pi}{2} < x < \frac{\pi}{2}$$ The points \(A\) and \(B\) are maximum points on \(C\).

1. Find the coordinates of \(B\) in terms of e. [5]

The finite region \(R\) lies between \(C\) and the line \(AB\).

1. Show that the area of \(R\) is $$\frac{2}{e} \arccos \left(\frac{1}{e}\right) + 2\ln \left(e + \sqrt{(e^2 - 1)}\right) - \frac{4}{e} \sqrt{(e^2 - 1)}.$$ [arccos \(x\) is an alternative notation for \(\cos^{-1} x\)] [8]

A curve has equation \(y = 2x^2 - 8x\sqrt{x} + 8x + 1\) for \(x \geq 0\)

1. Prove that the curve has a maximum point at \((1, 3)\) Fully justify your answer. [9 marks]
2. Find the coordinates of the other stationary point of the curve and state its nature. [2 marks]

Prove that the curve with equation $$y = 2x^5 + 5x^4 + 10x^3 - 8$$ has only one stationary point, stating its coordinates. [6 marks]

A curve has equation $$y = \frac{a}{\sqrt{x}} + bx^2 + \frac{c}{x^3} \quad \text{for } x > 0$$ where \(a\), \(b\) and \(c\) are positive constants. The curve has a single turning point. Use the second derivative of \(y\) to determine the nature of this turning point. You do not need to find the coordinates of the turning point. Fully justify your answer. [7 marks]

A curve has equation $$y = x^3 - 6x + \frac{9}{x}$$

1. Show that the \(x\) coordinates of the stationary points of the curve satisfy the equation $$x^4 - 2x^2 - 3 = 0$$ [3 marks]
2. Deduce that the curve has exactly two stationary points. [3 marks]
3. Find the coordinates and nature of the two stationary points. Fully justify your answer. [4 marks]
4. Write down the equation of a line which is a tangent to the curve in two places. [1 mark]

A continuous curve has equation \(y = f(x)\) The curve passes through the points \(A(2, 1)\), \(B(4, 5)\) and \(C(6, 1)\) It is given that \(f'(4) = 0\) Jasmin made two statements about the nature of the curve \(y = f(x)\) at the point \(B\): Statement 1: There is a turning point at \(B\) Statement 2: There is a maximum point at \(B\)

1. Draw a sketch of the curve \(y = f(x)\) such that Statement 1 is correct and Statement 2 is correct. [1 mark]
2. Draw a sketch of the curve \(y = f(x)\) such that Statement 1 is correct and Statement 2 is not correct. [1 mark]
3. Draw a sketch of the curve \(y = f(x)\) such that Statement 1 is not correct and Statement 2 is not correct. [1 mark]

It is given that for the continuous function \(g\) • \(g'(1) = 0\) • \(g'(4) = 0\) • \(g''(x) = 2x - 5\)

1. Determine the nature of each of the turning points of \(g\) Fully justify your answer. [3 marks]
2. Find the set of values of \(x\) for which \(g\) is an increasing function. [2 marks]

Rakti makes open-topped cylindrical planters out of thin sheets of galvanised steel. She bends a rectangle of steel to make an open cylinder and welds the joint. She then welds this cylinder to the circumference of a circular base. \includegraphics{figure_11} The planter must have a capacity of \(8000\text{cm}^3\) Welding is time consuming, so Rakti wants the total length of weld to be a minimum. Calculate the radius, \(r\), and height, \(h\), of a planter which requires the minimum total length of weld. Fully justify your answers, giving them to an appropriate degree of accuracy. [9 marks]

A curve has gradient function $$\frac{dy}{dx} = 3x^2 - 12x + c$$ The curve has a turning point at \((-1, 1)\)

1. Find the coordinates of the other turning point of the curve. Fully justify your answer. [6 marks]
2. Find the set of values of \(x\) for which \(y\) is increasing. [2 marks]

A fire crew is tackling a grass fire on horizontal ground. The crew directs a single jet of water which flows continuously from point \(A\). \includegraphics{figure_11} The path of the jet can be modelled by the equation $$y = -0.0125x^2 + 0.5x - 2.55$$ where \(x\) metres is the horizontal distance of the jet from the fire truck at \(O\) and \(y\) metres is the height of the jet above the ground. The coordinates of point \(A\) are \((a, 0)\)

1. 1. Find the value of \(a\). [3 marks]
   2. Find the horizontal distance from \(A\) to the point where the jet hits the ground. [1 mark]
2. Calculate the maximum vertical height reached by the jet. [4 marks]
3. A vertical wall is located 11 metres horizontally from \(A\) in the direction of the jet. The height of the wall is 2.3 metres. Using the model, determine whether the jet passes over the wall, stating any necessary modelling assumption. [3 marks]

A piece of wire of length 66 cm is bent to form the five sides of a pentagon. The pentagon consists of three sides of a rectangle and two sides of an equilateral triangle. The sides of the rectangle measure \(x\) cm and \(y\) cm and the sides of the triangle measure \(x\) cm, as shown in the diagram below. \includegraphics{figure_10}

1. 1. You are given that \(\sin 60° = \frac{\sqrt{3}}{2}\) Explain why the area of the triangle is \(\frac{\sqrt{3}}{4}x^2\) [1 mark]
   2. Show that the area enclosed by the wire, \(A\) cm\(^2\), can be expressed by the formula $$A = 33x - \frac{1}{4}(6 - \sqrt{3})x^2$$ [3 marks]
2. Use calculus to find the value of \(x\) for which the wire encloses the maximum area. Give your answer in the form \(p + q\sqrt{3}\), where \(p\) and \(q\) are integers. Fully justify your answer. [7 marks]

Prove that the graph of the curve with equation $$y = x^3 + 15x - \frac{18}{x}$$ has no stationary points. [5 marks]

Find the coordinates, in terms of \(a\), of the minimum point on the curve \(y = x^2 - 5x + a\), where \(a\) is a constant. Fully justify your answer. [3 marks]

A curve, C, has equation $$y = \frac{e^{3x-5}}{x^2}$$ Show that C has exactly one stationary point. Fully justify your answer. [7 marks]

An open-topped fish tank is to be made for an aquarium. It will have a square horizontal base, rectangular vertical sides and a volume of 60 m\(^3\) The materials cost:

• £15 per m\(^2\) for the base
• £8 per m\(^2\) for the sides.

1. Modelling the sides and base of the fish tank as laminae, use calculus to find the height of the tank for which the overall cost of the materials has its minimum value. Fully justify your answer. [8 marks]
2. 1. In reality, the thickness of the base and sides of the tank is 2.5 cm Briefly explain how you would refine your modelling to take account of the thickness of the sides and base of the tank. [1 mark]
   2. How would your refinement affect your answer to part (a)? [1 mark]

Find the coordinates of the stationary point of the curve with equation \((x + y - 2)^2 = e^y - 1\) [7 marks]

An elite athlete runs in a straight line to complete a 100-metre race. During the race, the athlete's velocity, \(v \text{ m s}^{-1}\), may be modelled by $$v = 11.71 - 11.68e^{-0.9t} - 0.03e^{0.3t}$$ where \(t\) is the time, in seconds, after the starting pistol is fired.

1. Find the maximum value of \(v\), giving your answer to one decimal place. Fully justify your answer. [8 marks]
2. Find an expression for the distance run in terms of \(t\). [6 marks]
3. The athlete's actual time for this race is 9.8 seconds. Comment on the accuracy of the model. [2 marks]

A cylinder is to be cut out of the circular face of a solid hemisphere. The cylinder and the hemisphere have the same axis of symmetry. The cylinder has height \(h\) and the hemisphere has a radius of \(R\). \includegraphics{figure_9}

1. Show that the volume, \(V\), of the cylinder is given by $$V = \pi R^2 h - \pi h^3$$ [3 marks]
2. Find the maximum volume of the cylinder in terms of \(R\). Fully justify your answer. [7 marks]

A gardener is creating flowerbeds in the shape of sectors of circles. The gardener uses an edging strip around the perimeter of each of the flowerbeds. The cost of the edging strip is £1.80 per metre and can be purchased for any length. One of the flowerbeds has a radius of 5 metres and an angle at the centre of 0.7 radians as shown in the diagram below. \includegraphics{figure_5}

1. 1. Find the area of this flowerbed. [2 marks]
   2. Find the cost of the edging strip required for this flowerbed. [3 marks]
2. A flowerbed is to be made with an area of 20 m²
   1. Show that the cost, £\(C\), of the edging strip required for this flowerbed is given by $$C = \frac{18}{5}\left(\frac{20}{r} + r\right)$$ where \(r\) is the radius measured in metres. [3 marks]
   2. Hence, show that the minimum cost of the edging strip for this flowerbed occurs when \(r \approx 4.5\) Fully justify your answer. [5 marks]