Math topic中文

Second derivative test

use of second derivative test to discriminate between maxima and minima

Practice questions

1The diagram shows a greenhouse built on a rectangular base. $140\text{ m}^2$ of polyethene is used to build the curved r…
2Find the gradient function of the curve $y = 2x^2 - \ln x + c$ . The curve touches the positive $x$ -axis at point $T$ . Fi…
3A cone of height, $h$ and radius, $r$ fits exactly inside a sphere of radius $5\text{ cm}$ as shown below. $\left(\text{Volume of cone} = \frac{1}{3}\pi r^2 h\right)$ …
4The diagram shows the glass test tube of radius $r\text{ cm}$ and height $(h+r)\text{ cm}$ with a hemisphere of radius $r\text{ cm}$ …
5A farmer uses 160 m of fencing to enclose a plot of his land in a shape that comprises an isosceles triangle and a recta…
6Solutions to this question by accurate drawing will not be accepted. In the trapezium $OABC$ , the point $A$ has coor…
7A factory manufactures miniature containers that are made up of a closed circular cylinder and a hemisphere which have t…
8[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…
9A 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…
10The diagram shows a prism in which the cross-section is a trapezium $ABCD$ . $AB = 4x\text{ cm}$ , $BC = DC = AD = 2x\text{ cm}$ …
11A miniature model in the shape of a triangular prism is shown above. A piece of wire, 36 cm in length, runs through all…
12A gardener plans to construct two flower beds, one a circle of radius $r$ m, the other a square of side $x$ m. To protec…
13A capsule is made of a cylinder with a hemisphere at one end of the cylinder. The cylinder has a height, $h\text{ cm}$ a…
14The diagram shows a paper weight which is made up of a solid hemisphere of radius $r$ cm, resting on top of a solid cyli…
15The point $(a, 2)$ is a stationary point on the graph $y = x^3 - x^2 - x + b$ , where $a$ and $b$ are positive constants.…
16The diagram shows a cat scratch stand which consists of a solid cylinder fixed to a solid cuboid. The cylinder has a rad…
17In $\Delta ABC$ , $\angle ABC = 90^\circ$ , $AB = 12\text{ cm}$ and $AC = 13\text{ cm}$ . The rectangle $BPQR$ is such that…
18求函数 $f(x) = x^3 - 3x^2 + 2$ 的极值点和极值。
19[The volume of a pyramid of height $h$ is $\tfrac{1}{3}\times(\text{Base Area})\times h$ .] The diagram shows a piece of…
20An 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…
21The equation of a curve is $y = x^2(6 - x)$ . Find the coordinates of the stationary points of the curve. Use the second…
22The figure shows the outline of a field, which consists of a semi-circular arc with radius $r\text{ m}$ and sides of a r…
23The equation of a curve is $y = \frac{\ln x}{x}$ . Find the coordinates of the stationary point of this curve. Leave your…
24Find the coordinates of the stationary points on the graph of $y = 2x^3 - 9x^2 + 13$ . Hence, determine the nature of the…
25The diagram shows a container, made of a thin sheet of plastic, in the shape of a hollow cylinder. The length of the con…
26The equation of a curve is $y = 2x^3 + 6x^2 - 18x + 5$ . Find the coordinates of the stationary points and determine the…
27The 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…
28The equation of a curve is $y = xe^{-3x}$ . Find the stationary point(s) and determine the nature of the stationary point…
29The 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…
30A particle moves in a straight line so that t seconds after leaving a fixed point O, its velocity v m/s, is given by $v = 16\bigl(2e^{-2t} - e^{-t}\bigr)$ …
31The diagram shows a shape made by cutting an equilateral triangle out of a rectangle of width $x$ cm. The perimeter of t…
32Peter constructed an open fish tank with a rectangular base of length $4l$ m, breadth $l$ m, and height $h$ m. He wanted…
33The diagram shows a solid object. It is made up of a cylinder with a hemisphere above it. The cylinder has a radius of $x\text{ cm}$ …
34 $ABCDEF$ is the cross-sectional plan of a temporary spectator stand, where $AB$ is parallel to $FC$ and $ED$ and $AE$ is…
35The figure below shows a right-angled triangular prism. The height and base of the triangular faces of the prism are $3x$ …
36The diagram shows a container which consists of an open cylinder of radius $r$ m and height $h$ m joined to a hemispheri…
37A farmer just bought a piece of land to grow his crops. This piece of land consists of an equilateral triangle of sides…
38The diagram shows the curve $y = \sin 2x + x$ for $0 \le x \le \pi$ radians. The point $M$ is the minimum point of the c…
39A curve is such that $\frac{dy}{dx} = \frac{8}{x^{2}} - 2.$ Given that the curve passes through the point (1, 5), find t…
40The equation of a curve is $y = \dfrac{\ln 2x}{x}$ . Find an expression for $\dfrac{d^{2}y}{dx^{2}}$ and show that $x\dfrac{d^{2}y}{dx^{2}} + 2\dfrac{dy}{dx} + \dfrac{1}{x^{2}} = 0$ …