CMR $a^{4}$ + $b^{4}$ + $c^{4}$ + $d^{4}$ ≥ 4abcd
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Question

CMR $a^{4}$ +  $b^{4}$ +  $c^{4}$ +  $d^{4}$  ≥ 4abcd
giúp em vs

hangbich · Answer

Giải thích các bước giải:
Ta có : $(x-y)^2\ge 0	o x^2+y^2\ge 2xy$
Ta có :$a^4+b^4\ge 2a^2b^2$ 
$c^4+d^4\ge 2c^2d^2$
$	o a^4+b^4+c^4+d^4\ge 2(a^2b^2+c^2d^2)=2((ab)^2+(cd)^2)\ge 2.2ab.cd=4abcd$
Dấu = xảy ra khi $a=b=c=d$

nghiemthutrang · Answer

Đáp án:
 chúc bạn học tốt
Giải thích các bước giải:
Ta có BĐT a+b&#x2265;2ab&#x21D4;(a+b)2&#x2265;(2ab)2" role="presentation" style="box-sizing: border-box; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0">a+b≥2$\sqrt{ab}$ 
a+b&#x2265;2ab&#x21D4;(a+b)2&#x2265;(2ab)2" role="presentation" style="box-sizing: border-box; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0">⇔(a+b)²≥(2$\sqrt{ab}$)² 
&#x21D4;a2+2ab+b2&#x2265;4ab&#x21D4;a2&#x2212;2ab+b2&#x2265;0&#x21D4;(a&#x2212;b)2&#x2265;0&#x2200;a,b" role="presentation" style="box-sizing: border-box; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0">⇔a²+2ab+b²≥4ab 
⇔a²−2ab+b²≥0
⇔a(a−b)²≥0∀a,b
Đẳng thức xảy ra khi (a&#x2212;b)2=0&#x21D2;a=b" role="presentation" style="box-sizing: border-box; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0">(a−b)²=0⇒a-b=0 ⇒a=b
Vậy ta có: a4+b4+c4+d4&#x2265;2a2b2+2c2d2=2[(ab)2+(cd)2]" role="presentation" style="box-sizing: border-box; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0">$a^{4}$ a4+b4&#x2265;2a4b4=2a2b2" role="presentation" style="box-sizing: border-box; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0">+$b^{4}$≥2$\sqrt{a^{4}.b^{4}}$ 
                 $c^{4}$ a4+b4&#x2265;2a4b4=2a2b2" role="presentation" style="box-sizing: border-box; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0">+$d^{4}$≥2$\sqrt{c^{4}.d^{4}}$ 
Cộng theo vế 2 BĐT trên ta có:
a4+b4+c4+d4&#x2265;2a2b2+2c2d2=2[(ab)2+(cd)2]" role="presentation" style="box-sizing: border-box; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0">$a^{4}$ a4+b4&#x2265;2a4b4=2a2b2" role="presentation" style="box-sizing: border-box; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0">+$b^{4}$+$c^{4}$ a4+b4&#x2265;2a4b4=2a2b2" role="presentation" style="box-sizing: border-box; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0">+$d^{4}$≥ a4+b4&#x2265;2a4b4=2a2b2" role="presentation" style="box-sizing: border-box; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0">2$\sqrt{a^{4}.b^{4}}$  + 2$\sqrt{c^{4}.d^{4}}$
Lại có: (ab)2+(cd)2&#x2265;2(ab)2(cd)2=2abcd" role="presentation" style="box-sizing: border-box; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0">(ab)²+(cd)²≥a4+b4+c4+d4&#x2265;2a2b2+2c2d2=2[(ab)2+(cd)2]" role="presentation" style="box-sizing: border-box; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0">a4+b4&#x2265;2a4b4=2a2b2" role="presentation" style="box-sizing: border-box; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0">2$\sqrt{(ab)^{2}.(cd)^{2}}$
&#x21D2;2[(ab)2+(cd)2]&#x2265;2&#x22C5;2abcd=4abcd" role="presentation" style="box-sizing: border-box; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0">⇒2[(ab)²+(cd)²]≥2⋅2abcd=4abcd
&#x21D2;VT=a4+b4+c4+d4&#x2265;4abcd=VP" role="presentation" style="box-sizing: border-box; display: inline-table; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0">⇒VT=a4+b4+c4+d4&#x2265;2a2b2+2c2d2=2[(ab)2+(cd)2]" role="presentation" style="box-sizing: border-box; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0">$a^{4}$ a4+b4&#x2265;2a4b4=2a2b2" role="presentation" style="box-sizing: border-box; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0">+$b^{4}$+$c^{4}$ a4+b4&#x2265;2a4b4=2a2b2" role="presentation" style="box-sizing: border-box; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 16px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0">+$d^{4}$ ≥4abcd=VP
Đẳng thức xảy ra khi$\left \{ {{a^{4}= b^{4} ; c^{4}=d^{4}} \atop {(ab)²=(cd)²}} \right.$ 
=> a=b=c=d

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CMR $a^{4}$ + $b^{4}$ + $c^{4}$ + $d^{4}$ ≥ 4abcd giúp em vs

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CMR a4a^{4} + b4b^{4} + c4c^{4} + d4d^{4} ≥ 4abcd giúp em vs

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CMR $a^{4}$ + $b^{4}$ + $c^{4}$ + $d^{4}$ ≥ 4abcd giúp em vs