Toán Lớp 8: 1)Tìm Min
a)A=x^2+x-2
b)B=x^2+y^2+x-6y+5
2)Tìm Max
b) B=12x-8y-4x^2-y^2+1
c)C=6x^2-x^2+3
d) D=2x-6y-x^2-y^2-2
Giúp e nhanh vs ạ
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Toán Lớp 8: 1)Tìm Min
a)A=x^2+x-2
b)B=x^2+y^2+x-6y+5
2)Tìm Max
b) B=12x-8y-4x^2-y^2+1
c)C=6x^2-x^2+3
d) D=2x-6y-x^2-y^2-2
Giúp e nhanh vs ạ
Comments ( 1 )
1)a)A = {x^2} + x – 2\\
= {x^2} + 2.x.\dfrac{1}{2} + \dfrac{1}{4} – \dfrac{9}{4}\\
= {\left( {x + \dfrac{1}{2}} \right)^2} – \dfrac{9}{4} \ge \dfrac{{ – 9}}{4}\\
\Leftrightarrow GTNN:A = – \dfrac{9}{4}\,khi:x = – \dfrac{1}{2}\\
b)B = {x^2} + {y^2} + x – 6y + 5\\
= {x^2} + x + {y^2} – 6y + 5\\
= {x^2} + 2.x.\dfrac{1}{2} + \dfrac{1}{4} + {y^2} – 6y + 9 – \dfrac{{17}}{4}\\
= {\left( {x + \dfrac{1}{2}} \right)^2} + {\left( {y – 3} \right)^2} – \dfrac{{17}}{4} \ge \dfrac{{ – 17}}{4}\\
\Leftrightarrow GTNN:B = – \dfrac{{17}}{4}\,khi:x = – \dfrac{1}{2};y = 3\\
2)\\
b)B = 12x – 8y – 4{x^2} – {y^2} + 1\\
= – \left( {4{x^2} – 12x} \right) – \left( {{y^2} + 8y} \right) + 1\\
= – \left( {4{x^2} – 12x + 9} \right) – \left( {{y^2} + 8y + 16} \right) + 9 + 16 + 1\\
= – {\left( {2x – 3} \right)^2} – {\left( {y + 4} \right)^2} + 26 \le 26\\
\Leftrightarrow GTLN:B = 26\,khi:\left\{ \begin{array}{l}
x = \dfrac{3}{2}\\
y = – 4
\end{array} \right.\\
c)C = 6{x^2} – {x^2} + 3\\
= 5{x^2} + 3 \ge 3\\
\Leftrightarrow GTNN:C = 3\,khi:x = 0\\
d)D = 2x – 6y – {x^2} – {y^2} – 2\\
= – \left( {{x^2} – 2x} \right) – \left( {{y^2} + 6y} \right) – 2\\
= – \left( {{x^2} – 2.x + 1} \right) – \left( {{y^2} + 6y + 9} \right) + 1 + 9 – 2\\
= – {\left( {x – 1} \right)^2} – {\left( {y + 3} \right)^2} + 8\\
\Leftrightarrow GTLN:D = 8\,khi:x = 1;y = – 3
\end{array}$