SUEC 高中 Adv Maths (Inequalities).pptx10. 𝒑𝒈 𝟕𝟒
4 + 𝑎2
− 4𝑎
= 𝑎2
−4𝑎 + 4
= 𝑎2 −4𝑎 + 22 − 22 + 4
= (𝑎 − 2)2
(𝑎 − 2)2
≥ 0
∴ 4 + 𝑎2
> 4𝑎
1 >
4𝑎
4 + 𝑎2
4𝑎
4 + 𝑎2 < 1
(𝑥 + 𝑦)
1
𝑥
+
1
𝑦
− 4
= 1 +
𝑥
𝑦
+
𝑦
𝑥
+ 1 − 4
=
𝑥
𝑦
+
𝑦
𝑥
− 2
= 𝑥2
+ 𝑦2
− 2𝑥𝑦
= 𝑥 − 2
(𝑥 − 𝑦)2≥ 0
∴ (𝑥 + 𝑦)
1
𝑥
+
1
𝑦
> 4
17. 𝒑𝒈 𝟕𝟒
𝑎
𝑏
+
𝑏
𝑐
+
𝑐
𝑎
≥ 3 − −①
𝑏
𝑎
+
𝑐
𝑏
+
𝑎
𝑐
≥ 3 − −②
① × ②
𝑎
𝑏
+
𝑏
𝑐
+
𝑐
𝑎
𝑏
𝑎
+
𝑐
𝑏
+
𝑎
𝑐
≥ 9
𝑥3 + 4
2𝑥2
2
≥
4
𝑥
𝑥6
+ 8𝑥3
+ 16
4𝑥4
≥
4
𝑥
𝑥7 + 8𝑥4 + 16𝑥 ≥ 16𝑥4
𝑥7
− 8𝑥4
+ 16𝑥 ≥ 0
𝑥(𝑥6 − 8𝑥3 + 16) ≥ 0
𝑥(𝑥3 − 4)2 ≥ 0
𝑥 ≥ 0 𝑜𝑟 𝑥 ≥
3
4
𝑥 +
4
𝑥2
≥
4
𝑥
𝑥 +
4
𝑥2
≥
4
5
4
𝑥 +
4
𝑥2
≥ 3
19. 𝒑𝒈 𝟕𝟒
𝑎2 + 𝑏2 + 𝑐2 + 𝑑2 ≥ 4 𝑎2𝑏2𝑐2𝑑2 − −①
1
𝑎2 +
1
𝑏2 +
1
𝑐2 +
1
𝑑2 ≥ 4
1
𝑎2𝑏2𝑐2𝑑2 − −②
① × ② 𝑎2 + 𝑏2 + 𝑐2 + 𝑑2
1
𝑎2
+
1
𝑏2
+
1
𝑐2
+
1
𝑑2
≥ 16
Editor's Notes https://www.mathsisfun.com/algebra/inequality-properties.html https://www.mathsisfun.com/algebra/inequality-properties.html
https://www.mathsisfun.com/algebra/inequality-properties.html
https://www.mathsisfun.com/algebra/inequality-properties.html