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k biếttt
+) \(E=x^2-6x+9+x^2-22x+121=2x^2-28x+130\)
\(\Rightarrow2E=4x^2-56x+242=\left(4x^2-56x+196\right)+46=\left(2x-14\right)^2+46\)
Vì \(\left(2x-14\right)^2\ge0\Rightarrow2E=\left(2x-14\right)^2+46\ge46\Rightarrow E\ge23\)
Dấu "=" xảy ra khi x=7
Vậy Emin=23 khi x=7
+) \(F=\frac{-2}{x^2-2x+5}=\frac{-2}{x^2-2x+1+4}=\frac{-2}{\left(x-1\right)^2+4}\)
Vì \(\left(x-1\right)^2\ge0\Rightarrow\left(x-1\right)^2+4\ge4\Rightarrow F=\frac{-2}{\left(x-1\right)^2+4}\le-\frac{2}{4}=-\frac{1}{2}\)
Dấu "=" xảy ra khi x=1
Vậy Fmin=-1/2 khi x=1
+) \(G=\left(x+1\right)\left(x-2\right)\left(x-3\right)\left(x-6\right)=\left(x^2-6x+x-6\right)\left(x^2-3x-2x+6\right)=\left(x^2-5x-6\right)\left(x^2-5x+6\right)\)
Đặt x2-5x=t, ta được:
\(G=\left(t-6\right)\left(t+6\right)=t^2-36=\left(x^2-5x\right)^2-36\)
Vì \(\left(x^2-5x\right)^2\ge0\Rightarrow G=\left(x^2-5x\right)^2-36\ge36\)
Dấu "=" xảy ra khi x=0 hoặc x=5
Vậy Gmin=36 khi x=0 hoặc x=5
3
Ta có: \(a^2+b^2+c^2\ge ab+bc+ca\Leftrightarrow a^2+b^2+c^2+2ab+2bc+2ca\ge3\left(ab+bc+ca\right)\)
\(\Leftrightarrow a^2+2a\left(b+c\right)+\left(b+c\right)^2\ge3\left(ab+bc+ca\right)\)
\(\Leftrightarrow\left(a+b+c\right)^2\ge3\left(ab+bc+ca\right)\Rightarrow\text{Đ}PCM\)
2b)
Ta có: \(x^2+y^2-4x-2y+5=0\Leftrightarrow x^2+y^2-4x-2y+4+1=0\Leftrightarrow\left(x-2\right)^2+\left(y-1\right)^2=0\)
\(\Leftrightarrow\hept{\begin{cases}\left(x-2\right)^2=0\\\left(y-1\right)^2=0\end{cases}\Leftrightarrow\hept{\begin{cases}x=2\\y=1\end{cases}}}\)
c) \(x^4-11x^2+4x-21=0\Leftrightarrow x^4-10x^2+25-x^2+4x-4=0\)
\(\Leftrightarrow\left(x^2-5\right)^2-\left(x-2\right)^2=0\Leftrightarrow\left(x^2-x-5+2\right)\left(x^2+x-5-2\right)=0\)
đến đây tự làm
\(B=\left(\frac{x-4}{x\left(x-2\right)}+\frac{2}{x-2}\right):\left(\frac{x+2}{x}-\frac{x}{x-2}\right)\)
\(< =>B=\left(\frac{x-4}{x\left(x-2\right)}+\frac{2x}{x\left(x-2\right)}\right):\left(\frac{\left(x-2\right)\left(x+2\right)}{x\left(x-2\right)}+\frac{x.x}{x\left(x-2\right)}\right)\)
\(< =>B=\left(\frac{x-4+2x}{x\left(x-2\right)}\right):\left(\frac{x^2-4}{x\left(x-2\right)}+\frac{x^2}{x\left(x-2\right)}\right)\)
\(< =>B=\frac{3x-4}{x\left(x-2\right)}:\frac{x^2-4+x^2}{x\left(x-2\right)}\)
\(< =>B=\frac{3x-4}{x\left(x-2\right)}.\frac{x\left(x-2\right)}{2x^2-4}\)
\(< =>B=\frac{3x-4}{2x^2-4}\)
\(b,\)Với \(x=-2\)thì
\(B=\frac{3\left(-2\right)-4}{2\left(-2\right)^2-4}=\frac{-6-4}{8-4}=-\frac{10}{4}=-\frac{5}{2}\)
\(ĐKXĐ:x\ne2;x\ne0\)
a
\(B=\left[\frac{x-4}{x\left(x-2\right)}+\frac{2}{x-2}\right]:\left(\frac{x+2}{x}-\frac{x}{x-2}\right)\)
\(=\frac{x-4+2x}{x\left(x-2\right)}:\frac{\left(x+2\right)\left(x-2\right)-x^2}{x\left(x-2\right)}\)
\(=\frac{3x-4}{x^2-4-x^2}=-\frac{3x-4}{4}\)
b
\(B=-\frac{3x-4}{4}=-\frac{3\cdot\left(-2\right)-4}{4}=\frac{5}{2}\)
c
\(\left|B\right|-2x=5\Leftrightarrow\left|B\right|=5+2x\)
\(B=-\frac{3x-4}{4}\Leftrightarrow-\frac{3x-4}{4}\ge0\Leftrightarrow x\le\frac{4}{3}\)
\(B=\frac{3x-4}{4}\Leftrightarrow x>\frac{4}{3}\)
Xét các trường hợp của x thì ra nghiệm bạn nhé
d
\(\left(2-x\right)B=-\frac{\left(2-x\right)\left(3x-4\right)}{4}\)
Để ( 2 - x ).B đạt giá trị nhỏ nhất thì ( 2 - x ) ( 3x - 4 ) đạt giá trị lớn nhất
Casio sẽ giúp chúng ta phần này
e
Để B là số nguyên âm lớn nhất hay \(B=-1\Leftrightarrow-\frac{3x-4}{4}=-1\Leftrightarrow x=\frac{8}{3}\)
g
\(\left|B\right|+3< 2x-1\)
Làm hệt như câu c nhé :D
\(A=x^2-3x+5\)
\(=x^2-3x+\frac{9}{4}+\frac{11}{4}\)
\(=\left(x-\frac{3}{2}\right)^2+\frac{11}{4}\)
\(\left(x-\frac{3}{2}\right)^2\ge0\Rightarrow A\ge\frac{11}{4}\)
Dấu "=" xảy ra khi \(x-\frac{3}{2}=0\Rightarrow x=\frac{3}{2}\)
Vậy Min A = \(\frac{11}{4}\Leftrightarrow x=\frac{3}{2}\)
a) \(A=x^2-3x+5\)
\("="\Leftrightarrow x=\frac{11}{4}\Rightarrow x=\frac{3}{2};\frac{11}{4}\)
b) \(B=\left(2x-1\right)^2+\left(x+2\right)^2\)
\("="\Leftrightarrow x=5\Rightarrow x=0;5\)
c) \(C=4x-x^2+3\)
\("="\Leftrightarrow x=7\Rightarrow x=2;7\)
d) \(D=x^4+x^2+2\)
\("="\Leftrightarrow x=2\Rightarrow x=0;2\)
B=a+a+a+\(\dfrac{27}{a^2}-\dfrac{22}{a^2}\)\(\ge\)\(3\sqrt[3]{\dfrac{a.a.27}{a^2}}+a-\dfrac{22}{a^2}\)\(\ge\)9+3-\(\dfrac{22}{9}\)=\(\dfrac{86}{9}\)
vậy MinB=\(\dfrac{86}{9}\) dấu bằng xảy ra khi a=3