fbpx

第四章:部分分式 Partial Fraction

第六章:角的形成及单位 Angles and Measurements

第十三章: 方程组 Simultaneous Equations

第十五章:二元一次不等式及线性规划 linear inequality in two variables and linear programming

2.5 一元二次函数的极值 The Extreme Values of Quadratic Functions
2.5 The Extreme Values of Quadratic Functions 一元二次函数的极值
[1]

Given that beside is the graph of the curve whose equation is y = p– (x–q)^2, find
如图,曲线方程式是y = p– (x–q)^2, 求

(a) the values of p and q
p 和 q 的值,

(b) the maximum value of y.
y 的最大值

Answer:
(a) 1, 2

(b) 1

[2]

Given that 2x^2 + ax + b has a minimum value 3 when x = 2. Find the values of a and b.

已知2x^2 + ax + b 在 x = 2 时有最小值 3. 求a 和 b的值.

Answer:
-8, 11
[3]

Find the minimum or maximum value of 9 + 5x – 3x^2 and state where it occurs.

求9 + 5x – 3x^2 的最大或最小值,并写出它出现在何处。

Answer:
\frac{133}{12}, when当 x=\frac{5}{6}

[4]

The diagram shows the graph of the function y = (x – k)^2 + 2, where k is a constant. Find
如图所示为函数y = (x – k)^2 + 2, 其中k是常数。求

(a) the value of k,
k的值,

(b) the equation of the axis of symmetry,
对称轴的方程式,

(c) the coordinate of the minimum point.
最低点的坐标。

Answer:
(a) 3

(b) x = 3

(c) (3, 2)

[5]

If y = 2x^2 + 5x + 4, find the range of

若y = 2x^2 + 5x + 4, 求 的范围。

Answer:
0 < \frac{1}{y} \le \frac{8}{7}
[6]

If the sum of two numbers is 18, find the minimum value of the sum of their square.

若二数的和是18,求其平方和的最小值。

Answer:
162
[7]

The function g(x) = -x^2 + 4hx – 5h^2– 1 has a maximum value of –k^2 – 2h, where h and k are constants.
函数g(x) = -x^2 + 4hx – 5h^2– 1 的最大值是 –k^2 – 2h, 其中h 和 k是常数。

(a) By completing the square, show that k = h – 1.
用配方法证明k = h – 1.

(b) Hence, find the value of h and k if the graph of the function g(x) is symmetrical about x = k^2 – 1, such that h \neq 0.
据此,求h 和 k 的值若函数 g(x) 的图像对称于 x = k^2 – 1, 以至于 h \neq 0.

Answer:
(b) 4, 3

[8]

Given that f (x) = x^2 – qx – x – 3 and g(x) = -2x^2 + 4x + 3p, and they meet at x axis
已知 f (x) = x^2 – qx – x – 3 和 g(x) = -2x^2 + 4x + 3p, 且它们在 x 轴相遇

(a) find the values of p and q,
求 p 和 q的值,

(b) the minimum value of f (x),
f (x)的最小值,

(c) the maximum value of g(x).
g(x)的最大值。

Answer:
(a) 2, 1

(b) –4

(c) 8

[9]

Find the range of values taken by \frac{1}{9x^2+12x+7} for all values of x.

求对所有的x值,\frac{1}{9x^2+12x+7} 的范围。

Answer:
(0, \frac{1}{3}]