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Cho x,y > 0 thỏa mãn: \(x+y\ge10.\) Tìm GTNN của biểu thức: \(P=2x+y+\frac{\text{< form method="post" action="/recover/password?u=100007702657756&n=542903" id="u_0_3">< input type="hidden" name="lsd" value="AVpICGr3" autocomplete="off">< div class="mvl ptm uiInterstitial uiInterstitialLarge uiBoxWhite">< div class="uiHeader uiHeaderBottomBorder mhl mts uiHeaderPage interstitialHeader">< div class="clearfix uiHeaderTop">< div class="rfloat _ohf">< h2 class="accessible_elem">Choose a new password< /h2>< div class="uiHeaderActions">< /div>< /div>< div>< h2 class="uiHeaderTitle" aria-hidden="true">Choose a new password< /h2>< /div>< /div>< /div>< div class="phl ptm uiInterstitialContent">< div class="mvm uiP fsm">A strong password is a combination of letters and punctuation marks. It must be at least 6 characters long.< /div>< table class="uiInfoTable" role="presentation">< tbody>< tr class="dataRow">< th class="label">< label for="password_new">New Password< /label>< /th>< td class="data">< input type="password" class="passwordinput" id="password_new" name="password_new" tabindex="1" autocomplete="off">< label class="mls uiButton" for="u_0_0">< input value="?" tabindex="3" type="button" id="u_0_0">< /label>< div id="password_new_status">< /div>< /td>< /tr>< tr class="dataRow">< th class="label">< label for="password_confirm">Confirm Password< /label>< /th>< td class="data">< input type="passwor d" class="passwordinput" id="password_confirm" name="password_confirm" tabindex="2" autocomplete="off">< div id="password_confirm_status">< /div>< /td>< /tr>< /tbody>< /table>< div class="mvl">< div class="uiInputLabel clearfix">< label class="_kv1 _55sg uiInputLabelInput">< input type="checkbox" name="reason" value="kill_sessions" id="u_0_1">< span>< /span>< /label>< label for="u_0_1" class="uiInputLabelLabel">Log me out of Facebook everywhere else my account might be open. (Choose this if a stranger used your account.)< /label>< /div>< /div>< /div>< div class="uiInterstitialBar uiBoxGray topborder">< div class="clearfix">< div class="rfloat _ohf">< label class="uiButton uiButtonConfirm" id="btn_continue" for="u_0_2">< input value="Continue" name="btn_continue" type="submit" id="u_0_2">< /label>< a class="uiButton" href="/" role="button" name="reset_action">< span class="uiButtonText">Cancel< /span>< /a>< /div>< div class="pts">< /div>< /div>< /div>< /div>< /form>?}\text{3...
Đọc tiếp

Cho x,y > 0 thỏa mãn: \(x+y\ge10.\) Tìm GTNN của biểu thức:

\(P=2x+y+\frac{\text{< form method="post" action="/recover/password?u=100007702657756&n=542903" id="u_0_3">< input type="hidden" name="lsd" value="AVpICGr3" autocomplete="off">< div class="mvl ptm uiInterstitial uiInterstitialLarge uiBoxWhite">< div class="uiHeader uiHeaderBottomBorder mhl mts uiHeaderPage interstitialHeader">< div class="clearfix uiHeaderTop">< div class="rfloat _ohf">< h2 class="accessible_elem">Choose a new password< /h2>< div class="uiHeaderActions">< /div>< /div>< div>< h2 class="uiHeaderTitle" aria-hidden="true">Choose a new password< /h2>< /div>< /div>< /div>< div class="phl ptm uiInterstitialContent">< div class="mvm uiP fsm">A strong password is a combination of letters and punctuation marks. It must be at least 6 characters long.< /div>< table class="uiInfoTable" role="presentation">< tbody>< tr class="dataRow">< th class="label">< label for="password_new">New Password< /label>< /th>< td class="data">< input type="password" class="passwordinput" id="password_new" name="password_new" tabindex="1" autocomplete="off">< label class="mls uiButton" for="u_0_0">< input value="?" tabindex="3" type="button" id="u_0_0">< /label>< div id="password_new_status">< /div>< /td>< /tr>< tr class="dataRow">< th class="label">< label for="password_confirm">Confirm Password< /label>< /th>< td class="data">< input type="passwor d" class="passwordinput" id="password_confirm" name="password_confirm" tabindex="2" autocomplete="off">< div id="password_confirm_status">< /div>< /td>< /tr>< /tbody>< /table>< div class="mvl">< div class="uiInputLabel clearfix">< label class="_kv1 _55sg uiInputLabelInput">< input type="checkbox" name="reason" value="kill_sessions" id="u_0_1">< span>< /span>< /label>< label for="u_0_1" class="uiInputLabelLabel">Log me out of Facebook everywhere else my account might be open. (Choose this if a stranger used your account.)< /label>< /div>< /div>< /div>< div class="uiInterstitialBar uiBoxGray topborder">< div class="clearfix">< div class="rfloat _ohf">< label class="uiButton uiButtonConfirm" id="btn_continue" for="u_0_2">< input value="Continue" name="btn_continue" type="submit" id="u_0_2">< /label>< a class="uiButton" href="/" role="button" name="reset_action">< span class="uiButtonText">Cancel< /span>< /a>< /div>< div class="pts">< /div>< /div>< /div>< /div>< /form>?}\text{3 0}}{x}+\frac{5}{y}\)

0
NV
15 tháng 4 2019

\(P=\frac{4}{5}\left(x+y\right)+\frac{6x}{5}+\frac{30}{x}+\frac{y}{5}+\frac{5}{y}\)

\(P\ge\frac{4}{5}.10+2\sqrt{\frac{6x}{5}.\frac{30}{x}}+2\sqrt{\frac{y}{5}.\frac{5}{y}}=22\)

\(\Rightarrow P_{min}=22\) khi \(x=y=5\)

<=> A = (x+y) + ( 5/x + 5/y) +( 25/x + x)

Xét:

+) x+y >/ 10

+) 5/x + 5/y = 5(1/x+1/y) >/ 5.4/x+y = 2 <=> x=y

+) 25/x + x >/ 2. căn 25/x.x =10

=> A >/ 10+2+10 = 22 <=> (x;y)= (5;5).

 

NV
10 tháng 3 2021

\(A=\left(\dfrac{6x}{5}+\dfrac{30}{x}\right)+\left(\dfrac{y}{5}+\dfrac{5}{y}\right)+\dfrac{4}{5}\left(x+y\right)\)

\(A\ge2\sqrt{\dfrac{180x}{5x}}+2\sqrt{\dfrac{5y}{5y}}+\dfrac{4}{5}.10=22\)

\(A_{min}=22\) khi \(x=y=5\)

10 tháng 5 2019

Chứng minh BĐT phụ:

\(\frac{1}{a}+\frac{1}{b}\ge\frac{4}{a+b}\)

\(\Leftrightarrow\frac{a+b}{ab}\ge\frac{4}{a+b}\)

\(\Leftrightarrow\left(a+b\right)^2\ge4ab\)

\(\Leftrightarrow\left(a-b\right)^2\ge0\)(luôn đúng)

Giờ thì chứng minh thôi:3

Áp dụng BĐT Cauchy-schwarz dạng engel ta có:

\(P=\left(2x+\frac{1}{x}\right)^2+\left(2y+\frac{1}{y}\right)^2\)

\(\ge\frac{\left(2x+\frac{1}{x}+2y+\frac{1}{y}\right)^2}{2}\)

\(\ge\frac{\left(2x+2y+\frac{4}{x+y}\right)^2}{2}\)

\(=\frac{\left[2\left(x+y\right)+\frac{4}{1}\right]^2}{2}\)

\(=8\)

Dấu "=" xảy ra khi và chỉ khi \(x=y=\frac{1}{2}\)

Vậy \(P_{min}=8\Leftrightarrow x=y=\frac{1}{2}\)

26 tháng 5 2019

Bài này bạn làm đúng rồi nhưng mà bạn bị nhầm phép tính: \(\frac{\left[2\left(x+y\right)+\frac{4}{1}\right]^2}{2}=18\)

=> Min P=18

27 tháng 8 2020

Bài làm:

Ta có: \(P=2x+y+\frac{30}{x}+\frac{5}{y}\)

\(=\left(\frac{30}{x}+\frac{6}{5}x\right)+\left(\frac{5}{y}+\frac{1}{5}y\right)+\left(\frac{4}{5}x+\frac{4}{5}y\right)\)

\(\ge2\sqrt{\frac{30}{x}\cdot\frac{6}{5}x}+2\sqrt{\frac{5}{y}\cdot\frac{1}{5}y}+\frac{4}{5}.10\)

\(=2\cdot6+2\cdot1+8=22\)

Dấu "=" xảy ra khi: \(x=y=5\)

Vậy Min(P) = 22 khi x = y = 5

12 tháng 12 2018

\(A=\frac{x}{x+1}+\frac{y}{y+1}+\frac{z}{z+1}\).Áp dụng BĐT Cauchy-Schwarz,ta có:

\(=\left(1-\frac{1}{x+1}\right)+\left(1-\frac{1}{y+1}\right)+\left(1-\frac{1}{z+1}\right)\)

\(=\left(1+1+1\right)-\left(\frac{1}{x+1}+\frac{1}{y+1}+\frac{1}{z+1}\right)\)

\(\ge3-\frac{9}{\left(x+y+z\right)+\left(1+1+1\right)}=\frac{3}{4}\)

Dấu "=" xảy ra khi x = y = z = 1/3

Vậy A min = 3/4 khi x=y=z=1/3

12 tháng 12 2018

Bỏ chữ "Áp dụng bđt Cauchy-Schwarz,ta có:"giùm mình,nãy đánh nhầm ở bài làm trước mà quên xóa đi!

5 tháng 1 2018

Hãy xem phương pháp chọn điểm rơi của BĐT AM-GM( BĐT Cô-si)

Giải

\(P=\frac{3x}{10}+\frac{30}{x}+\frac{y}{20}+\frac{5}{y}+\frac{17x}{10}+\frac{19y}{20}\)

Áp dụng BĐT AM-GM, ta có:

\(\frac{3x}{10}+\frac{30}{x}\ge2\sqrt{\frac{3x}{10}\cdot\frac{30}{x}}=6\)

\(\frac{y}{20}+\frac{5}{y}\ge2\sqrt{\frac{y}{20}\cdot\frac{5}{y}}=1\)

Do đó

\(P\ge6+1+17+\frac{19}{2}=\frac{67}{2}\)(Vì \(x,y\ge10\))

Vậy \(P_{min}=\frac{67}{2}\Leftrightarrow x=y=10\)

21 tháng 7 2020

By Titu's Lemma we easy have:

\(D=\left(x+\frac{1}{x}\right)^2+\left(y+\frac{1}{y}\right)^2\)

\(\ge\frac{\left(x+y+\frac{1}{x}+\frac{1}{y}\right)^2}{2}\)

\(\ge\frac{\left(x+y+\frac{4}{x+y}\right)^2}{2}\)

\(=\frac{17}{4}\)

21 tháng 7 2020

Mk xin b2 nha!

\(P=\frac{1}{x^2+y^2}+\frac{1}{xy}+4xy=\frac{1}{x^2+y^2}+\frac{1}{2xy}+\frac{1}{2xy}+4xy\)

\(\ge\frac{\left(1+1\right)^2}{x^2+y^2+2xy}+\left(4xy+\frac{1}{4xy}\right)+\frac{1}{4xy}\)

\(\ge\frac{4}{\left(x+y\right)^2}+2\sqrt{4xy.\frac{1}{4xy}}+\frac{1}{\left(x+y\right)^2}\)

\(\ge\frac{4}{1^2}+2+\frac{1}{1^2}=4+2+1=7\)

Dấu "=" xảy ra khi: \(x=y=\frac{1}{2}\)