1.07n Stationary points: find maxima, minima using derivatives

A function f is defined for all real values of \(x\) as $$f(x) = x^4 + 5x^3$$ The function has exactly two stationary points when \(x = 0\) and \(x = -\frac{15}{4}\)

1. 1. Find \(f''(x)\) [2 marks]
   2. Determine the nature of the stationary points. Fully justify your answer. [4 marks]
2. State the range of values of \(x\) for which $$f(x) = x^4 + 5x^3$$ is an increasing function. [1 mark]
3. A second function g is defined for all real values of \(x\) as $$g(x) = x^4 - 5x^3$$
   1. State the single transformation which maps f onto g. [1 mark]
   2. State the range of values of \(x\) for which g is an increasing function. [1 mark]

The function f is defined by $$f(x) = \frac{x^2 + 10}{2x + 5}$$ where f has its maximum possible domain. The curve \(y = f(x)\) intersects the line \(y = x\) at the points P and Q as shown below. \includegraphics{figure_10}

1. State the value of \(x\) which is not in the domain of f. [1 mark]
2. Explain how you know that the function f is many-to-one. [2 marks]
3. 1. Show that the \(x\)-coordinates of P and Q satisfy the equation $$x^2 + 5x - 10 = 0$$ [2 marks]
   2. Hence, find the exact \(x\)-coordinate of P and the exact \(x\)-coordinate of Q. [1 mark]
4. Show that P and Q are stationary points of the curve. Fully justify your answer. [5 marks]
5. Using set notation, state the range of f. [2 marks]

A new design for a company logo is to be made from two sectors of a circle, \(ORP\) and \(OQS\), and a rhombus \(OSTR\), as shown in the diagram below. \includegraphics{figure_7} The points \(P\), \(O\) and \(Q\) lie on a straight line and the angle \(ROS\) is \(\theta\) radians. A large copy of the logo, with \(PQ = 5\) metres, is to be put on a wall.

1. Show that the area of the logo, \(A\) square metres, is given by $$A = \frac{25}{8}(\pi - \theta + 2\sin\theta)$$ [4 marks]
2. 1. Show that the maximum value of \(A\) occurs when \(\theta = \frac{\pi}{3}\) Fully justify your answer. [6 marks]
   2. Find the exact maximum value of \(A\) [2 marks]
3. Without further calculation, state how your answers to parts (b)(i) and (b)(ii) would change if \(PQ\) were increased to 10 metres. [2 marks]

\includegraphics{figure_4} Figure 4 shows the plan view of the design for a swimming pool. The shape of this pool \(ABCDEA\) consists of a rectangular section \(ABDE\) joined to a semicircular section \(BCD\) as shown in Figure 4. Given that \(AE = 2x\) metres, \(ED = y\) metres and the area of the pool is \(250\text{m}^2\),

1. show that the perimeter, \(P\) metres, of the pool is given by $$P = 2x + \frac{250}{x} + \frac{\pi x}{2}$$ [4]
2. Explain why \(0 < x < \sqrt{\frac{500}{\pi}}\) [2]
3. Find the minimum perimeter of the pool, giving your answer to \(3\) significant figures. [4]

\includegraphics{figure_2} Figure 2 shows a solid cuboid \(ABCDEFGH\). \(AB = x\) cm, \(BC = 2x\) cm, \(AE = h\) cm The total surface area of the cuboid is 180 cm\(^2\). The volume of the cuboid is \(V\) cm\(^3\).

1. Show that \(V = 60x - \frac{4x^3}{3}\) [4]

Given that \(x\) can vary,

1. use calculus to find, to 3 significant figures, the value of \(x\) for which \(V\) is a maximum. Justify that this value of \(x\) gives a maximum value of \(V\). [5]
2. Find the maximum value of \(V\), giving your answer to the nearest cm\(^3\). [2]

Sketch the following curves.

1. \(y = \frac{2}{x}\) [2]
2. \(y = x^3 - 6x^2 + 9x\) [5]

The equation of a curve is $$y = 6x^4 + 8x^3 - 21x^2 + 12x - 6.$$

1. In this question you must show detailed reasoning. Determine
   [12]
2. On the axes in the Printed Answer Booklet, sketch the curve whose equation is $$y = 6x^4 + 8x^3 - 21x^2 + 12x - 6.$$ [3]

The function g is defined by $$g(x) = \mathrm{e}^{\sin x} \quad (0 \leq x \leq 2\pi)$$ The diagram below shows the graph of \(y = g(x)\) \includegraphics{figure_8}

1. Find the \(x\)-coordinate of each of the stationary points of the graph of \(y = g(x)\), giving your answers in exact form. [1 mark]
2. Use Simpson's rule with 3 ordinates to estimate $$\int_0^\pi g(x) \, \mathrm{d}x$$ giving your answer to two decimal places. [3 marks]
3. Explain how Simpson's rule could be used to find a more accurate estimate of the integral in part (b). [1 mark]

A curve \(C\) has equation \(y = \frac{1}{9}x^3 - kx + 5\). A point \(Q\) lies on \(C\) and is such that the tangent to \(C\) at \(Q\) has gradient \(-9\). The \(x\)-coordinate of \(Q\) is \(3\).

1. Show that \(k = 12\). [3]
2. Find the coordinates of each of the stationary points of \(C\) and determine their nature. [6]
3. Sketch the curve \(C\), clearly labelling the stationary points and the point where the curve crosses the \(y\)-axis. [2]

A curve \(C\) has equation \(f(x) = 3x^3 - 5x^2 + x - 6\).

1. Find the coordinates of the stationary points of \(C\) and determine their nature. [8]
2. Without solving the equations, determine the number of distinct real roots for each of the following:
   1. \(3x^3 - 5x^2 + x + 1 = 0\),
   2. \(6x^3 - 10x^2 + 2x + 1 = 0\). [4]

A curve C has equation \(y = -x^3 + 12x - 20\).

1. Find the coordinates of the stationary points of C and determine their nature. [7]
2. Determine the range of values of \(x\) for which the curve is decreasing. Give your answer in set notation. [3]

The diagram below shows a closed box in the form of a cuboid, which is such that the length of its base is twice the width of its base. The volume of the box is 9000 cm³. The total surface area of the box is denoted by \(S\) cm². \includegraphics{figure_14}

1. Show that \(S = 4x^2 + \frac{27000}{x}\), where \(x\) cm denotes the width of the base. [3]
2. Find the minimum value of \(S\), showing that the value you have found is a minimum value. [5]

The curve \(y = ax^4 + bx^3 + 18x^2\) has a point of inflection at \((1, 11)\).

1. Show that \(2a + b + 6 = 0\). [2]
2. Find the values of the constants \(a\) and \(b\) and show that the curve has another point of inflection at \((3, 27)\). [8]
3. Sketch the curve, identifying all the stationary points including their nature. [6]

\includegraphics{figure_5} Figure 5 shows a sketch of the curve \(C\) with equation \(y = (x - 2)^2(x + 3)\) The region \(R\), shown shaded in Figure 5, is bounded by \(C\), the vertical line passing through the maximum turning point of \(C\) and the \(x\)-axis. Find the exact area of \(R\). *(Solutions based entirely on graphical or numerical methods are not acceptable.)* [6]

A sculpture formed from a prism is fixed on a horizontal platform, as shown in the diagram. The shape of the cross-section of the sculpture can be modelled by the equation \(x^2 + 2xy + 2y^2 = 10\), where \(x\) and \(y\) are measured in metres. The \(x\) and \(y\) axes are horizontal and vertical respectively. \includegraphics{figure_3} Find the maximum vertical height above the platform of the sculpture. [8 marks]

A curve has equation \(y = g(x)\). Given that • \(g(x)\) is a cubic expression in which the coefficient of \(x^3\) is equal to the coefficient of \(x\) • the curve with equation \(y = g(x)\) passes through the origin • the curve with equation \(y = g(x)\) has a stationary point at \((2, 9)\)

1. find \(g(x)\), [7]
2. prove that the stationary point at \((2, 9)\) is a maximum. [2]

\includegraphics{figure_5} Figure 5 shows a sketch of the curve \(C\) with equation \(y = f(x)\). The curve \(C\) crosses the \(x\)-axis at the origin, \(O\), and at the points \(A\) and \(B\) as shown in Figure 5. Given that $$f'(x) = k - 4x - 3x^2$$ where \(k\) is constant.

1. show that \(C\) has a point of inflection at \(x = -\frac{2}{3}\) [3] Given also that the distance \(AB = 4\sqrt{2}\)
2. find, showing your working, the integer value of \(k\). [5]

A curve is defined for \(x \geq 0\) by the equation $$y = 6x - 2x^{\frac{1}{2}}$$

1. Find \(\frac{dy}{dx}\). [2 marks]
2. The curve has one stationary point. Find the coordinates of the stationary point and determine whether it is a maximum or minimum point. Fully justify your answer. [3 marks]

A particle \(P\) moves in a straight line. At time \(t\) s the displacement \(s\) cm from a fixed point \(O\) is given by: $$s = \frac{1}{6}\left(8t^3 - 105t^2 + 144t + 540\right).$$ Find the distance between the points at which the particle is instantaneously at rest. [7]

In this question you must show detailed algebraic reasoning. Find the coordinates of any stationary points on the curve below. $$y = (1 - 3x)(3 - x)^3$$ [5]

\includegraphics{figure_10} The figure above shows solid right prism of height \(h\) cm. The cross section of the prism is a circular sector of radius \(r\) cm, subtending an angle of 2 radians at the centre.

1. Given that the volume of the prism is 1000 cm\(^3\), show clearly that $$S = 2r^2 + \frac{4000}{r},$$ where \(S\) cm\(^2\) is the total surface area of the prism. [5]
2. Hence determine the value of \(r\) and the value of \(h\) which make \(S\) least, fully justifying your answer. [7]