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BÀI 1:
a) \(x^4+2x^2y+y^2=\left(x^2+y\right)^2\)
b) \(\left(2a+b\right)^2-\left(2b+a\right)^2=\left(2a+b+2b+a\right)\left(2a+b-2b-a\right)\)
\(=\left(3a+3b\right)\left(a-b\right)=3\left(a+b\right)\left(a-b\right)\)
c) \(\left(a^3-b^3\right)+\left(a-b\right)^2=\left(a-b\right)\left(a^2+ab+b^2\right)+\left(a-b\right)^2\)
\(=\left(a-b\right)\left[a^2+ab+b^2+\left(a-b\right)\right]=\left(a-b\right)\left(a^2+ab+b^2+a-b\right)\)
d) \(\left(x^2+1\right)^2-4x^2=\left(x^2+1-2x\right)\left(x^2+1+2x\right)=\left(x-1\right)^2\left(x+1\right)^2\)
e) \(\left(y^3+8\right)+\left(y^2-4\right)=\left(y+2\right)\left(y^2-y+2\right)\)
f) \(1-\left(x^2-2xy+y^2\right)=1-\left(x-y\right)^2=\left(1-x+y\right)\left(1+x-y\right)\)
g) \(x^4-1=\left(x-1\right)\left(x+1\right)\left(x^2+1\right)\)
h) ktra lại đề
m) \(\left(x-a\right)^4-\left(x+a\right)^4=-8ax\left(a^2+x^2\right)\)
Bài 1:
a) |x - 1,38| + |2y - 4,2| ≥ 0 ∀ x; y. Dấu '=' xảy ra khi x = 1,38; y = 2,1
Vậy ....
b) |x - y| + |y + \(\frac{9}{25}\)| ≥ 0 ∀ x,y. Dấu "=" xảy ra khi x=y=\(-\frac{9}{25}\)
Vậy ...
Bài 2:
a) 2|3x - 2| - 1 ≥ -1 ∀x. Vậy Amin= -1 khi x=\(\frac{2}{3}\)
b) x2 + 3|y - 2| - 1 ≥ -1 ∀x,y. Vậy Bmin = -1 khi x = 0; y = 2
c) C = |x + 32| + |54 - x| ≥ |x + 32 + 54 - x| = 86.
Cmin = 86 khi (x + 32)(54 - x) ≥ 0 ⇔ -32 ≤ x ≤ 54
d) |5x - 2| + |3y + 12| ≥ 0 ∀x,y. ⇒ 4 - |5x - 2| - |3y + 12| ≤ 4
Vậy Dmax = 4 khi x = \(\frac{2}{5}\); y = -4
Bài 1 quan trong là đoán dấu đẳng thức.
1/ Có: \(36=\left(3+2+1\right)\left(a^2+b^2+c^2\right)\ge\left(\sqrt{3}a+\sqrt{2}b+c\right)^2\)
\(\therefore\sqrt{3}a+\sqrt{2}b+c\le6\)
\(\frac{1}{3}\left(\frac{a}{bc}+\frac{3b}{2ca}\right)+\frac{3}{2}\left(\frac{b}{ca}+\frac{2c}{ab}\right)+2\left(\frac{c}{ab}+\frac{a}{3bc}\right)\)
\(\ge\frac{\sqrt{6}}{3c}+\frac{3\sqrt{2}}{a}+\frac{4\sqrt{3}}{3b}\)
\(=\frac{\left(\frac{\sqrt{6}}{3}\right)}{c}+\frac{\left(3\sqrt{6}\right)}{\sqrt{3}a}+\frac{\left(\frac{4\sqrt{6}}{3}\right)}{\sqrt{2}b}\)
\(\ge\frac{\left(\sqrt{\frac{\sqrt{6}}{3}}+\sqrt{3\sqrt{6}}+\sqrt{\frac{4\sqrt{6}}{3}}\right)^2}{\sqrt{3}a+\sqrt{2}b+c}\ge2\sqrt{6}\)
Đẳng thức xảy ra khi \(a=\sqrt{3},b=\sqrt{2},c=1\)
a) M = y + 2; b) M = 2 ( a – b ) 3 .