Question

cho x + y = 2. Tìm giá trị nhỏ nhất của biểu thức : S = x² + y²

Answer 1

(x&#x2212;y)2&#x2265;0&lt;=&gt;x2+y2&#x2265;2xy&lt;=&gt;2x2+2y2&#x2265;x2+y2+2xy&lt;=&gt;2(x2+y2)&#x2265;(x+y)2=22=4&lt;=&gt;x2+y2&#x2265;2" role="presentation" style="border:0px; color:rgb(40, 40, 40); direction:ltr; display:inline-table; float:none; font-family:helvea,arial,sans-serif; font-size:16.38px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:45.202em; overflow-wrap:normal; padding:1px 0px; position:relative; text-align:center; white-space:nowrap; width:10000em" class="MathJax_CHTML mjx-chtml mjx-full-width">2 (x−y)2≥0<=>x2+y2≥2xy<=>2x2+2y2≥x2+y2+2xy<=>2(x2+y2)≥(x+y)2=22=4<=>x2+y2≥2(x−y)2≥0<=>x2+y2≥2xy<=>2x2+2y2≥x2+y2+2xy<=>2(x2+y2)≥(x+y)2=22=4<=>x2+y2≥2

Answer 2

(x2+y2)(1+1)&#x2265;(x+y)2&#x21D2;(x2+y2)2&#x2265;4&#x21D2;x2+y2&#x2265;2" role="presentation" style="border:0px; direction:ltr; display:inline-block; float:none; font-size:16.38px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:1px 0px; position:relative; white-space:nowrap; word-spacing:normal" class="MathJax_CHTML mjx-chtml">(x2+y2)(1+1)&#x2265;(x+y)2&#x21D2;(x2+y2)2&#x2265;4&#x21D2;x2+y2&#x2265;2" role="presentation" style="border:0px; direction:ltr; display:inline-block; float:none; font-size:16.38px; line-height:0; margin:0px; max-height:none; max-width:none; min-height:0px; min-width:0px; overflow-wrap:normal; padding:1px 0px; position:relative; white-space:nowrap; word-spacing:normal" class="MathJax_CHTML mjx-chtml">hc tốt

Answer 3

Từ x + y = 2. Ta có : y = 2- x

Do đó : S = x² - ( 2 - x )² 

               = 2 ( x - 1 )² + 2 > 2

Vậy min S = 2 \(\Leftrightarrow x=y=1\)