Ta có : \(ab+bc+ac\le a^2+b^2+c^2\Leftrightarrow2\left(ab+bc+ac\right)\le2\left(a^2+b^2+c^2\right)\Leftrightarrow\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2\ge0\)
Vì BĐT cuối luôn đúng nên ta có : \(a^2+b^2+c^2\ge ab+bc+ac\)
Theo Bất đẳng thức tam giác ta có :
\(a< b+c\Rightarrow a.a< a\left(b+c\right)\Leftrightarrow a^2< ab+ac\) (1)
\(b< a+c\Rightarrow b.b< b\left(a+c\right)\Leftrightarrow b^2< ab+bc\)(2)
\(c< a+b\Rightarrow c.c< c\left(a+b\right)\Leftrightarrow c^2< ac+bc\)(3)
Cộng (1) , (2) , (3) theo vế ta được : \(a^2+b^2+c^2< 2\left(ab+bc+ac\right)\)
Từ đó suy ra đpcm
Nguyễn Thị Ngọc Ánh k lm thì biến đừng hòng kiếm
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a<1" role="presentation" style="border:0px; direction:ltr; display:inline-block; float:none; font-size:18.06px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; padding:1px 0px; position:relative; white-space:nowrap; word-spacing:normal; word-wrap:normal" class="MathJax_CHTML mjx-chtml">
b<1,c<1" role="presentation" style="border:0px; direction:ltr; display:inline-block; float:none; font-size:18.06px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; padding:1px 0px; position:relative; white-space:nowrap; word-spacing:normal; word-wrap:normal" class="MathJax_CHTML mjx-chtml">
(1−a)(1−b)(1−c)>0" role="presentation" style="border:0px; direction:ltr; display:inline-block; float:none; font-size:18.06px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; padding:1px 0px; position:relative; white-space:nowrap; word-spacing:normal; word-wrap:normal" class="MathJax_CHTML mjx-chtml">
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1–(a+b+c)+ab+bc+ca>abc" role="presentation" style="border:0px; direction:ltr; display:inline-table; float:none; font-size:18.06px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; padding:1px 0px; position:relative; white-space:nowrap; word-spacing:normal; word-wrap:normal" class="MathJax_CHTML mjx-chtml">
abc<−1+ab+bc+ca" role="presentation" style="border:0px; direction:ltr; display:inline-block; float:none; font-size:18.06px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; padding:1px 0px; position:relative; white-space:nowrap; word-spacing:normal; word-wrap:normal" class="MathJax_CHTML mjx-chtml">
2abc<−2+2ab+2bc+2ca" role="presentation" style="border:0px; direction:ltr; display:inline-block; float:none; font-size:18.06px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; padding:1px 0px; position:relative; white-space:nowrap; word-spacing:normal; word-wrap:normal" class="MathJax_CHTML mjx-chtml">
a2+b2+c2+2abc<a2+b2+c2–2+2ab+2bc+2ca" role="presentation" style="border:0px; direction:ltr; display:inline-block; float:none; font-size:18.06px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; padding:1px 0px; position:relative; white-space:nowrap; word-spacing:normal; word-wrap:normal" class="MathJax_CHTML mjx-chtml">
Ta có:
a<b+c
--> a+a<a+b+c
--> 2a<22a<2
--> a<1a<1
Tương tự ta có : b<1,c<1b<1,c<1
Suy ra: (1−a)(1−b)(1−c)>0(1−a)(1−b)(1−c)>0
⇔ (1–b–a+ab)(1–c)>0(1–b–a+ab)(1–c)>0
⇔ 1–c–b+bc–a+ac+ab–abc>01–c–b+bc–a+ac+ab–abc>0
⇔ 1–(a+b+c)+ab+bc+ca>abc1–(a+b+c)+ab+bc+ca>abc
Nên abc<−1+ab+bc+caabc<−1+ab+bc+ca
⇔ 2abc<−2+2ab+2bc+2ca2abc<−2+2ab+2bc+2ca
⇔ a2+b2+
K phải kiếm trên :CHUYÊN ĐỀ 2: Bất đẳng thức. Các bài toán tìm giá trị lớn nhất , nhỏ nhất. pptx là có k nhé . Tự nghĩ mk sẽ k
Ta có: a2 +b2 +c2 - ab + bc + ca.0)()()(.21222 accbba
Đẳng thức xảy ra khi và chỉ khi a = b = c.
Vậy: ab + bc + caa2 +b2 +c2. Lại có: a < b + c a2 < a.(b + c) (1)
Tương tự: b2 < b.(a + c) (2) ,c2 < c.(b + a) (3).
a2 +b2 +c2 < a.(b + c) + b.(a + c) + c.(b + a) = 2.(ab + bc + ca).
Ta có:
a<b+ca<b+c
--> a+a<a+b+ca+a<a+b+c
--> 2a<22a<2
--> a<1a<1
Tương tự ta có : b<1,c<1b<1,c<1
Suy ra: (1−a)(1−b)(1−c)>0(1−a)(1−b)(1−c)>0
⇔ (1–b–a+ab)(1–c)>0(1–b–a+ab)(1–c)>0
⇔ 1–c–b+bc–a+ac+ab–abc>01–c–b+bc–a+ac+ab–abc>0
⇔ 1–(a+b+c)+ab+bc+ca>abc1–(a+b+c)+ab+bc+ca>abc
Nên abc<−1+ab+bc+caabc<−1+ab+bc+ca
⇔ 2abc<−2+2ab+2bc+2ca2abc<−2+2ab+2bc+2ca
⇔ a2+b2+
