Math topic中文

Stationary points

stationary points (maximum and minimum turning points and stationary points of inflexion)

Practice questions

1The curve $C$ has equation $y = \frac{x^2 + x + 3}{x - 2}$ . Without using a calculator, find the set of values that $y$ …
2The diagram shows a greenhouse built on a rectangular base. $140\text{ m}^2$ of polyethene is used to build the curved r…
3The equation of a curve is given by $y = x + \sin^2 x$ for $0 \le x \le \pi$ . Write down the expression for $\frac{\mathrm{d}y}{\mathrm{d}x}$ …
4A curve has equation $y = \sqrt{ax + (bx)^2} + cx^3 + d$ where $a$ , $b$ , $c$ and $d$ are constants. The curve cuts the x…
5A cone of height, $h$ and radius, $r$ fits exactly inside a sphere of radius $5\text{ cm}$ as shown below. $\left(\text{Volume of cone} = \frac{1}{3}\pi r^2 h\right)$ …
6The diagram shows the glass test tube of radius $r\text{ cm}$ and height $(h+r)\text{ cm}$ with a hemisphere of radius $r\text{ cm}$ …
7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. $1$
8 $ABCD$ is a rectangle which fits inside a semicircle of radius 8 cm and centre $O$ . It is given that $AB = x$ cm and $BC = y$ …
9A farmer uses 160 m of fencing to enclose a plot of his land in a shape that comprises an isosceles triangle and a recta…
10Solutions to this question by accurate drawing will not be accepted. In the trapezium $OABC$ , the point $A$ has coor…
11A factory manufactures miniature containers that are made up of a closed circular cylinder and a hemisphere which have t…
12The equation of a curve is $y = e^{-2x}\tan x$ . Show that $\dfrac{dy}{dx} = e^{-2x}(1 - \tan x)^2$ . Find, in terms of $\pi$ …
13A curve has the equation $y = x^3 + kx^2 + kx + 8$ . Find the set of values of $k$ such that the curve is an increasing f…
14[The volume of a sphere of radius r is $\dfrac{4}{3}\pi r^3$ , and its surface area is $4\pi r^2$ .] The diagram shows a b…
15A curve is defined by the equation $y = \ln\left(\frac{\text{e}^x}{x^2 + 10}\right)$ . Show that the curve has no turning…
16The diagram below shows part of the graph of $y = \frac{x^2 + 2x + 5}{x + 3}$ . Find the coordinates of the minimum point…
17A cone of height 12 cm and base radius 5 cm is placed over a cylinder of radius $r$ cm and height $h$ cm. The cone is in…
18The diagram shows a prism in which the cross-section is a trapezium $ABCD$ . $AB = 4x\text{ cm}$ , $BC = DC = AD = 2x\text{ cm}$ …
19画出曲线 $y = x^3 - 4x$ 的草图，标出零点、极值点和拐点坐标。
20Given that $y = x(2x-4)^3$ . find the coordinates of the stationary points of the curve, determine the nature of each of…
21A miniature model in the shape of a triangular prism is shown above. A piece of wire, 36 cm in length, runs through all…
22The diagram shows part of the curve $y = \sqrt{4x + 5}$ . The normal of the curve at $P$ meets the $x$ -axis at $T$ . The $x$ …
23A particle travels in a straight line so that, $t$ seconds after leaving a fixed point $O$ , its velocity, $v\text{ m s}^{-1}$ …
24A gardener plans to construct two flower beds, one a circle of radius $r$ m, the other a square of side $x$ m. To protec…
25A curve has the equation $y = x^3 - 3x^2 + 3x - 7$ . Find the coordinates of the stationary point on the curve. Observe t…
26The diagram shows part of the curve $y = 2 \sin 2x + 1$ . $P$ is the maximum point of the curve and $Q$ is the point on t…
27A cyclist moves along a straight road from home $O$ . His velocity, $v\text{ m/min}$ , is given by $v = 25 + 10t - 3t^2$ ,…
28A curve has the equation $y = 3(4 - x)^{3} + 5$ . Find the coordinates of the stationary point of the curve. Determine th…
29A capsule is made of a cylinder with a hemisphere at one end of the cylinder. The cylinder has a height, $h\text{ cm}$ a…
30The diagram shows a glass window $ABCDEF$ , consisting of a rectangle $ABEF$ of height $3x\text{ cm}$ and width $y\text{ cm}$ …
31The diagram shows part of the curve $y = 27 + \frac{8}{(x-2)^3}$ which crosses the axes at points $A$ and $B$ . Explain w…
32The diagram shows a paper weight which is made up of a solid hemisphere of radius $r$ cm, resting on top of a solid cyli…
33The point $(a, 2)$ is a stationary point on the graph $y = x^3 - x^2 - x + b$ , where $a$ and $b$ are positive constants.…
34By using a suitable substitution, or otherwise, solve the equation $3\log_6 x = \log_x 3 + \log_x 2 + \log_5 25.$ Sket…
35已知函数 $f(x)=e^{x-k}-x$ ， $(x∈R)(1)$ 当 $k=0$ 时，若函数 $f(x)\geqslant m$ 在 $R$ 上恒成立，求实数 $m$ 的取值范围； (2)试判断当 $k > 1$ 时，函数 $f(x)$ …
36A curve has the equation $y = \frac{e^{2x}}{x-2}$ , where $x > 2$ . Find $\dfrac{dy}{dx}$ . Find the exact value of the coo…
37A particle moving in a straight line, passes through a fixed point $O$ with a velocity of $4\text{ m/s}$ . Its accelerati…
38The diagram shows a cat scratch stand which consists of a solid cylinder fixed to a solid cuboid. The cylinder has a rad…
39已知函数 $f(x)= \dfrac {ax}{e^{x}}$ 在 $x=0$ 处的切线方程为 $y=x$ ． (1)求 $a$ 的值； (2)若对任意的 $x∈(0,2)$ ，都有 $\dfrac {1}{f(x)} > k+2x-x^{2}$ …
40The equation of a curve is $y = \frac{x}{e^{2x}}$ . Find an expression for $\frac{dy}{dx}$ . Given that $x$ is changing at…
41In $\Delta ABC$ , $\angle ABC = 90^\circ$ , $AB = 12\text{ cm}$ and $AC = 13\text{ cm}$ . The rectangle $BPQR$ is such that…
42The diagram shows a lake (shaded) bordered by a straight level road AXB. Point C is on the surface of the lake and B is…
43已知函数 $f(x)=m\ln x$ ． (1)讨论函数 $F(x)=f(x)+ \dfrac {1}{x}-1$ 的单调性； (2)定义：“对于在区域 $D$ 上有定义的函数 $y=f(x)$ 和 $y=g(x)$ ，若满足 $f(x)\leqslant g(x)$ …
44求函数 $f(x) = x^3 - 3x^2 + 2$ 的极值点和极值。
45[The volume of a pyramid of height $h$ is $\tfrac{1}{3}\times(\text{Base Area})\times h$ .] The diagram shows a piece of…
46An architect uses 40 cm of wire to design 2 window frames for a room model. He wants as much light as possible to pass t…
47The equation of a curve is $y = x^2(6 - x)$ . Find the coordinates of the stationary points of the curve. Use the second…
48The figure shows the outline of a field, which consists of a semi-circular arc with radius $r\text{ m}$ and sides of a r…
49The equation of a curve is $y = \frac{\ln x}{x}$ . Find the coordinates of the stationary point of this curve. Leave your…
50Find the coordinates of the stationary points on the graph of $y = 2x^3 - 9x^2 + 13$ . Hence, determine the nature of the…
51The diagram shows a solid object made from joining two hemispheres of radius $x$ cm to a cylinder of height $y$ cm. The…
52The equation of a curve is $y = e^{x^2 - 5x}$ . Find the expressions for $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$ . Find th…
53A particle $X$ travels in a straight line so that $t$ seconds after leaving a fixed point $O$ , its velocity $v$ m/s, is…
54The diagram shows a container, made of a thin sheet of plastic, in the shape of a hollow cylinder. The length of the con…
55The equation of a curve is $y = 2x^3 + 6x^2 - 18x + 5$ . Find the coordinates of the stationary points and determine the…
56The diagram shows parts of the graphs of $3y = 7x + 6$ and $x + 5y = 12$ . $T$ is a point on $3y = 7x + 6$ and $U$ is a p…
57The equation of a curve is $y = xe^{-3x}$ . Find the stationary point(s) and determine the nature of the stationary point…
58The diagram shows the junction of two corridors of width 9 m and 4 m which are at right angles. P and Q are variable poi…
59A particle moves in a straight line so that t seconds after leaving a fixed point O, its velocity v m/s, is given by $v = 16\bigl(2e^{-2t} - e^{-t}\bigr)$ …
60The diagram shows a shape made by cutting an equilateral triangle out of a rectangle of width $x$ cm. The perimeter of t…