Math topic中文

Nature stationary points

Determining the nature of the stationary points (local maximum and minimum points and points of inflexion) analytically, in simple cases, using the first derivative test or the second derivative test.

Practice questions

1[The volume of a cone of base radius $r$ and height $h$ is given by $V = \frac{1}{3}\pi r^2 h$ .] A manufacturer makes a…
2The curve C has equation $x + y = (x - y)^2$ . It is given that C has only one turning point. Show that $1 - \frac{dy}{dx} = \frac{2}{1 + 2x - 2y}$ …
3The curve C has equation $x + y = (x - y)^2.$ It is given that C has only one turning point. Show that $1 - \dfrac{dy}{dx} = \dfrac{2}{1 + 2x - 2y}$ …
4A curve $C_1$ has the equation $p^2x^2 - y^2 = p^2$ where $p > 1$ . Sketch $C_1$ , stating the coordinates of any points o…
5A sector of angle $\theta$ radians and radius $4\pi\text{ cm}$ is used to form the curved surface of a right circular co…
6As part of an Art Project, a student designed a lantern in the shape of an octahedron as shown in the diagram. The frame…
7The following diagram shows the dimensions of a trapezoidal prism with fixed volume $4k\sqrt{3}$ units $^{3}$ , with varia…
8The curve $C$ has equation $y = \frac{x - 2}{kx^2 + x - 2}$ , where $k > 1$ . Find the equation of the tangent at the poin…
9The graph of $y = \text{f}(x)$ has a maximum turning point at $(2,3)$ and passes through the origin. The lines $x = -1$ …
10The curve $C$ has equation $y = \frac{x^2 + kx + 4}{x + 1}$ , where $k$ is a constant. Find the range of values of $k$ if…
11A curve C has equation $y = k(x+1) + \dfrac{4}{x+1}$ , $x \in \mathbb{R}$ , $x \ne -1$ , where $0 < k < 2$ is a constant. T…
12A Hwa Chong student taking part in Project Day needs to design a rectangular poster to showcase his project. According t…
13[The surface area and volume of a sphere are given by $4\pi r^2$ and $\dfrac{4}{3}\pi r^3$ respectively.] An aquarist wh…
14Fig. 1 shows a piece of card, $ABCD$ , in the form of a rhombus of sides $a$ cm. The card is made up of 2 identical equil…
15[Cambridge/J78/II/6] An isosceles triangle $ABC$ with $AB = AC$ is inscribed in a fixed circle, centre $O$ , whose ra…
16Let f(x) be a cubic polynomial where f(2) = 6 and f(-1) = 15. Also, f(x) takes on stationary values at x = -2 and x = 1.…
17A Dunman High alumnus plans to donate money to build two koi ponds in the middle of Zheng Xin Yuan. The ponds are in the…
18In the diagram, point $Q$ is $400\text{ m}$ vertically above point $A$ . Initially, a balloon is at point $Q$ and moves h…
19The diagram below shows a parabola which cuts the $x$ -axis at $(-2a, 0)$ , $(2a, 0)$ and has a maximum point at $(0, 4a^2)$ …
20The diagram shows a shot put being projected with a velocity $v\text{ ms}^{-1}$ from the point $O$ at an angle $\theta$ …
21The following diagram shows the graphs of two differentiable functions $y=f(x)$ and $y=g(x)$ . At $x = c$ , the distance b…
22In a particular building, there is an $L$ -shaped corridor (see diagram below). Corridor 1 has a fixed width of 3 m and C…
23The cross section PQRS of a water passageway of depth d is as shown in the figure below. It is given that $PQ+QR+RS=2a$ …
24函数 $f(x)= \dfrac {x^{4}}{4}- \dfrac {x^{3}}{3}$ 的极值点为 $()$ A. $0$ B. $-1$ C. $0$ 或 $1$ D. $1$
25In this question, $a$ , $b$ , $c$ and $d$ are non-zero constants. The curve $C$ has equation $y = \frac{ax^2 + bx - 7}{x - c}$ …
26The equation of a curve is $y=\dfrac{x^2-3a^2}{x-a}$ , where $a$ is a positive constant. Find the equations of the asympt…
27A curve C has equation $y = f(x)$ , where $f(x) = \frac{2(x-1)^2}{x-3}$ . Sketch C, showing clearly the equations of the a…
28It is given that $\mathrm{f}(x) = x + \frac{m^2}{x-2}$ , where $0 < m < 1$ . Sketch the graph of $y = \mathrm{f}(x)$ , show…
29It is given that $f(x) = \begin{cases} \tfrac{1}{2}x^2, & \text{for } 1 \le x < 2, \\ \tfrac{1}{4}(-3x+14), & \text{for } 2 \le x < 4, \end{cases}$ …
30Fig. 1 shows the net of a triangular prism cut from a square cardboard of side length 20 cm. The net consists of one rec…
31A farmer has a plot of the land, $OAB$ , with adjacent sides $OA$ and $OB$ . With reference to $O$ as the origin, $OA$ and…
32A curve $C$ has equation $y = 2ax + \frac{b}{x}$ where $a$ and $b$ are non-zero real constants and $x \ne 0$ . Using diff…
33In the isosceles triangle $ABC$ , $AC = BC$ , $AB = 20\text{ cm}$ and $\angle BAC = 30^\circ$ . A rectangle $PQRS$ is inscr…
34It is given that $\mathrm{f}(x) = ax^3 + bx^2 + cx + d$ , where $a$ , $b$ , $c$ and $d$ are constants. The curve $C$ with e…
35[A sphere of radius $r$ has surface area $4\pi r^2$ and volume $\frac{4}{3}\pi r^3$ .] A water fountain is to be construc…
36The equation of a curve $y = \mathrm{f}(x)$ is in the form of a cubic polynomial. The curve has a maximum point at $(-1, 0)$ …
37画出曲线 $y = x^3 - 4x$ 的草图，标出零点、极值点和拐点坐标。
38Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\mathrm{f} : x \to \frac{4}{(x-4)^2}, \quad x \in \mathbb{R}, x \ne 4,$ …
39(Q2) [Cambridge/9231/2007] The curve C has equation $y = \frac{ax^2 + bx + c}{x + 4},$ where $a$ , $b$ and $c$ are cons…
40The function f is defined by $f : x \mapsto 4 + \dfrac{1}{(x-1)^2 + 1},\quad x \ge 0$ . The function g is defined by g :…
41A curve has equation $y=f(x)$ , where $f(x)=x^{3}+ax^{2}+bx+c$ . The curve $y=f(x)$ has a maximum point at $(1,-32)$ , whil…
42It is given that the curve $C$ has equation $y = \dfrac{-4x^2 - 6x - 11}{2x + 2}$ , $x \in \mathbb{R}$ , $x \ne -1$ . Witho…
43The curve $C$ has equation $y = \frac{x^2 + 4x + \lambda - 5}{x + r},$ where $\lambda$ is a non-zero constant, and a v…
44Projectile motion is the motion of a launched object, subject to only the constant acceleration of gravity, $g\text{ ms}^{-2}$ …
45The diagram shows the graph of $y = \text{f}'(x)$ , which passes through the origin and has turning points at $(2, 5)$ an…
46A curve $C$ has equation $y = \frac{2x^2 - 5ax + 2a^2 + 1}{x - a}$ , $x \ne a$ . Find the equations of the asymptotes, giv…
47The following diagram shows the dimensions of a trapezoidal prism with fixed volume $4k\sqrt{3}$ units $^3$ , with variabl…
48The curve C is defined by the parametric equations $x = a\left(1 + \dfrac{1}{t}\right)$ and $y = a\left(t - \dfrac{1}{t^2}\right)$ …
49A piece of vanguard sheet, $ABCDEF$ , is in the form of a regular hexagon of side $a\text{ cm}$ . A kite shape is cut out…
50MI PU2 Promo 9758/2019/02/Q1 Town A, Town B and Town C are located in Wakandi Country. The distance between Town A and T…
51A curve $C$ has equation $y = \frac{e^{kx}}{e^{kx} - e^{-kx}}$ , where $x \ne 0$ and $k \in \mathbb{R},\ k \ne 0$ . Show t…
52The curve $C$ has equation $y = \frac{2x^2 - 10x + 17}{x - 3}, \quad x \in \mathbb{R}, \quad x \ne 3$ . Without using a c…
53A company manufactures a closed container made of glass as shown in the figure below. The closed container, of negligibl…
54The curve $C$ has equation $y = \frac{3x^2 + ax + 3}{x + 1}$ , where $a$ is a constant, $a \ne 6$ . Find the set of values…
55Fig. 1 shows the net of a right regular hexagon-based prism cut from a rectangular cardboard of one side length 30 cm. T…
56The graph of $y = \frac{ax^2 + bx + c}{x+d}$ has a vertical asymptote with equation $x = -1$ and an oblique asymptote wi…
57A popular toy company is designing a new water play feature for children. The toy consists of a cylindrical water contai…
58A curve $G$ has equation $y = ax + b + \dfrac{c}{(x+1)^2}$ , where $a$ , $b$ and $c$ are positive integers. Let $R$ be the…
59In an experiment to test the design theory of launching a new space telescope into space, scientists monitor the movemen…
60A lighting consultant intends to construct a giant light structure for Christmas that is octahedral in shape. The edges…