
柯西不等式测验
Authored by 仲連 柯
Mathematics
11th Grade

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10 questions
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1.
MULTIPLE CHOICE QUESTION
30 sec • 1 pt
问题1: 如果a和b是实数,使用柯西不等式证明|ab| ≤ (a^2 + b^2)/2。
|ab| = a + b
|ab| ≤ (a^2 + b^2)/2
|ab| ≥ (a^2 + b^2)/2
|ab| = a^2 - b^2
2.
MULTIPLE CHOICE QUESTION
30 sec • 1 pt
问题2: 在给定正实数a和b的情况下,使用柯西不等式找到a和b的最小值。
a*b
a-b
a+b
a^2 + b^2
3.
MULTIPLE CHOICE QUESTION
30 sec • 1 pt
问题3: 如果(x1, y1), (x2, y2), ..., (xn, yn)是n个实数对,使用柯西不等式证明(∑(xi * yi))^2 ≤ (∑(xi^2))(∑(yi^2))。
∑(xi * yi) = (∑xi^2)(∑yi^2)
证明过程
∑(xi * yi) = (∑xi)^2 + (∑yi)^2
∑(xi * yi) = (∑xi)(∑yi)
4.
MULTIPLE CHOICE QUESTION
30 sec • 1 pt
问题4: 给定实数a, b, c,使用柯西不等式找到a, b, c的最大值。
√(a^2 + b^2 + c^2)
a^2 + b^2 + c^2
a * b * c
a + b + c
5.
MULTIPLE CHOICE QUESTION
30 sec • 1 pt
问题5: 如果(x1, x2, ..., xn)和(y1, y2, ..., yn)是n个实数对,使用柯西不等式证明|∑(xi * yi)| ≤ √(∑(xi^2) * ∑(yi^2))。
|∑(xi * yi)| ≥ √(∑(xi^2) * ∑(yi^2))
|∑(xi * yi)| = √(∑(xi^2) * ∑(yi^2))
|∑(xi * yi)| ≤ √(∑(xi^2) * ∑(yi^2))
|∑(xi * yi)| > √(∑(xi^2) * ∑(yi^2))
6.
MULTIPLE CHOICE QUESTION
30 sec • 1 pt
问题6: 在给定正实数a, b, c的情况下,使用柯西不等式找到a, b, c的最大值。
a^2 + b^2 + c^2
√(a^2 + b^2 + c^2)
a + b + c
a * b * c
7.
MULTIPLE CHOICE QUESTION
30 sec • 1 pt
问题7: 如果(x1, y1), (x2, y2), ..., (xn, yn)是n个实数对,使用柯西不等式证明(∑(xi * yi))^2 ≤ n * (∑(xi^2))(∑(yi^2))。
(∑(xi * yi))^2 ≤ n * (∑(xi^2))(∑(yi^2))
(∑(xi * yi))^2 < n * (∑(xi^2))(∑(yi^2))
(∑(xi * yi))^2 = n * (∑(xi^2))(∑(yi^2))
(∑(xi * yi))^2 ≥ n * (∑(xi^2))(∑(yi^2))
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