Toán Lớp 8: Tìm giá trị lớn nhất:
A= 5x- $x^{2}$
B= x- $x^{2}$
C= 4x- $x^{2}$ + 3
D= – $x^{2}$ + 6x – 11
E= 5 – 8x – $x^{2}$
Tìm giá trị nhỏ nhất
A= $x^{2}$ – 6x + 11
B= $x^{2}$ – 20x + 101
C= $x^{2}$ – 2x + $y^{2}$ +4y + 8
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B1)\\
A = 5x – {x^2} = \\
= – \left( {{x^2} – 5x} \right)\\
= – \left( {{x^2} – 2.x.\dfrac{5}{2} + \dfrac{{25}}{4}} \right) + \dfrac{{25}}{4}\\
= – {\left( {x – \dfrac{5}{2}} \right)^2} + \dfrac{{25}}{4} \le \dfrac{{25}}{4}\,\\
\Leftrightarrow GTLN:A = \dfrac{{25}}{4}\,khi:x = \dfrac{5}{2}\\
B = x – {x^2}\\
= – \left( {{x^2} – x} \right)\\
= – \left( {{x^2} – 2.x.\dfrac{1}{2} + \dfrac{1}{4}} \right) + \dfrac{1}{4}\\
= – {\left( {x – \dfrac{1}{2}} \right)^2} + \dfrac{1}{4} \le \dfrac{1}{4}\\
\Leftrightarrow GTLN:B = \dfrac{1}{4}\,khi:x = \dfrac{1}{2}\\
C = 4x – {x^2} + 3\\
= – \left( {{x^2} – 4x + 4} \right) + 4 + 3\\
= – {\left( {x – 2} \right)^2} + 7 \le 7\\
\Leftrightarrow GTLN:C = 7\,khi:x = 2\\
D = – {x^2} + 6x – 11\\
= – \left( {{x^2} – 6x + 9} \right) + 9 – 11\\
= – {\left( {x – 3} \right)^2} – 2 \le – 2\\
\Leftrightarrow GTLN:D = – 2\,khi:x = 3\\
E = 5 – 8x – {x^2}\\
= – \left( {{x^2} + 8x + 16} \right) + 16 + 5\\
= – {\left( {x + 4} \right)^2} + 21 \le 21\\
\Leftrightarrow GTLN:E = 21\,khi:x = – 4\\
B2)\\
A = {x^2} – 6x + 11\\
= {x^2} – 6x + 9 + 2\\
= {\left( {x – 3} \right)^2} + 2 \ge 2\\
\Leftrightarrow GTNN:A = 2\,khi:x = 3\\
B = {x^2} – 20x + 101\\
= {x^2} – 20x + 100 + 1\\
= {\left( {x – 10} \right)^2} + 1 \ge 1\\
\Leftrightarrow GTNN:B = 1\,khi:x = 10\\
C = {x^2} – 2x + {y^2} + 4y + 8\\
= {x^2} – 2x + 1 + {y^2} + 4y + 4 + 3\\
= {\left( {x – 1} \right)^2} + {\left( {y + 2} \right)^2} + 3 \ge 3\\
\Leftrightarrow GTNN:C = 3\,khi:\left\{ \begin{array}{l}
x = 1\\
y = – 2
\end{array} \right.
\end{array}$