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câu 1) ta có : \(M=\left(x^2-x\right)^2+\left(2x-1\right)^2=x^4-2x^3+x^2+4x^2-4x+1\)
\(=\left(x^2-x+2\right)^2-3=\left(\left(x-\dfrac{1}{2}\right)^2+\dfrac{7}{4}\right)^2-3\)
\(\Rightarrow\dfrac{1}{16}\le M\le61\)
\(\Rightarrow M_{min}=\dfrac{1}{16}\)khi \(x=\dfrac{1}{2}\) ; \(M_{max}=61\) khi \(x=3\)
câu 2) điều kiện xác định : \(0\le x\le2\)
đặt \(\sqrt{2x-x^2}=t\left(t\ge0\right)\)
\(\Rightarrow M=-t^2+4t+3=-\left(t-2\right)^2+7\)
\(\Rightarrow3\le M\le7\)
\(\Rightarrow M_{min}=3\)khi \(x=0\) ; \(M_{max}=7\) khi \(x=2\)câu 3) ta có : \(M=\left(x-2\right)^2+6\left|x-2\right|-6\ge-6\)
\(\Rightarrow M_{min}=-6\) khi \(x=2\)
4) điều kiện xác định \(-6\le x\le10\)
ta có : \(M=5\sqrt{x+6}+2\sqrt{10-x}-2\)
áp dụng bunhiacopxki dạng căn ta có :
\(-\sqrt{\left(5^2+2^2\right)\left(x+6+10-x\right)}\le5\sqrt{x+6}+2\sqrt{10-x}\le\sqrt{\left(5^2+2^2\right)\left(x+6+10-x\right)}\)
\(\Leftrightarrow-4\sqrt{29}\le5\sqrt{x+6}+2\sqrt{10-x}\le4\sqrt{29}\)
\(\Rightarrow-2-4\sqrt{29}\le B\le-2+4\sqrt{29}\)
\(\Rightarrow M_{max}=-2+4\sqrt{29}\) khi \(\dfrac{\sqrt{x+6}}{5}=\dfrac{\sqrt{10-x}}{2}\Leftrightarrow x=\dfrac{226}{29}\)
\(\Rightarrow M_{min}=-2-4\sqrt{29}\) dấu của bđt này o xảy ra câu 5 lm tương tự
\(A=\frac{3}{4}.4.x^2\left(8-x^2\right)\le\frac{3}{4}\left(x^2+8-x^2\right)^2=48\)
\(A_{max}=48\) khi \(x^2=8-x^2\Rightarrow x=\pm2\)
\(B=\frac{1}{2}\left(2x-1\right)\left(6-2x\right)\le\frac{1}{8}\left(2x-1+6-2x\right)^2=\frac{25}{8}\)
\(B_{max}=\frac{25}{8}\) khi \(2x-1=6-2x\Rightarrow x=\frac{7}{4}\)
\(C=\frac{1}{\sqrt{3}}.\sqrt{3}x\left(3-\sqrt{3}x\right)\le\frac{1}{4\sqrt{3}}\left(\sqrt{3}x+3-\sqrt{3}x\right)^2=\frac{3\sqrt{3}}{4}\)
\(C_{max}=\frac{3\sqrt{3}}{4}\) khi \(\sqrt{3}x=3-\sqrt{3}x=\frac{\sqrt{3}}{2}\)
\(D=\frac{1}{20}.20x\left(32-20x\right)\le\frac{1}{80}\left(20x+32-20x\right)^2=\frac{64}{5}\)
\(D_{max}=\frac{64}{5}\) khi \(20x=32-20x\Rightarrow x=\frac{4}{5}\)
\(E=\frac{4}{5}\left(5x-5\right)\left(8-5x\right)\le\frac{1}{5}\left(5x-5+8-5x\right)=\frac{9}{5}\)
\(E_{max}=\frac{9}{5}\) khi \(5x-5=8-5x\Leftrightarrow x=\frac{13}{10}\)
a/ \(\left\{{}\begin{matrix}\left(x-1\right)^2\ge0\\\left(y-5\right)^2\ge0\\\left(x-y+4\right)^2\ge0\end{matrix}\right.\) \(\Rightarrow\left(x-1\right)^2+\left(y-5\right)^2+\left(x-y+4\right)^2\ge0\)
\(A_{min}=0\) khi \(\left\{{}\begin{matrix}x=1\\y=5\end{matrix}\right.\)
b/ \(B=x^2y^2-6xy+9+x^2+4x+4-16\)
\(B=\left(xy-3\right)^2+\left(x+2\right)^2-16\ge-16\)
\(B_{min}=-16\) khi \(\left\{{}\begin{matrix}x=-2\\y=-\frac{3}{2}\end{matrix}\right.\)
c/ \(C=x^2+\frac{y^2}{4}+16+xy+8x+4y+\frac{59}{4}y^2-3y+2001\)
\(C=\left(x+\frac{y}{2}+4\right)^2+\frac{59}{4}\left(y-\frac{6}{59}\right)^2+\frac{118050}{59}\ge\frac{118050}{59}\)
\(C_{min}=\frac{118050}{59}\)
d/ \(D=\left(x^2-2x\right)\left(y^2+6y\right)+12\left(x^2-2x\right)+3\left(y^2+6y\right)+36\)
\(=\left(x^2-2x\right)\left(y^2+6y+12\right)+3\left(y^2+6y+12\right)\)
\(=\left(x^2-2x+3\right)\left(y^2+6y+12\right)\)
\(=\left[\left(x-1\right)^2+2\right]\left[\left(y+3\right)^2+3\right]\ge2.3=6\)
\(D_{min}=6\)
e/ \(E=a^2+\frac{b^2}{4}+\frac{9}{4}+ab-3a-\frac{3b}{2}+\frac{3b^2}{4}-\frac{3b}{2}+2014-\frac{9}{4}\)
\(=\left(a+\frac{b}{2}-\frac{3}{2}\right)^2+\frac{3}{4}\left(y-1\right)^2+2011\ge2011\)
\(E_{min}=2011\)
a:
\(A=\left|x-2013\right|+\left|2014-x\right|>=\left|x-2013+2014-x\right|=1\)
Dấu = xảy ra khi 2013<=x<=2014
\(B=\left|x-123\right|+\left|456-x\right|>=\left|x-123+456-x\right|=333\)
Dấu = xảy ra khi 123<=x<=456
b: \(\left|x\right|+2004>=2004\)
=>A<=2013/2004
Dấu = xảy ra khi x=0
\(B=\dfrac{\left|x\right|+2002+1}{\left|x\right|+2002}=1+\dfrac{1}{\left|x\right|+2002}< =1+\dfrac{1}{2002}=\dfrac{2003}{2002}\)
Dấu = xảy ra khi x=0
H = x(x+1)(x+2)(x+3)
=x(x+3)(x+1)(x+2)
=(x2+3x)(x2+3x+2)
Đặt t=x2+3x ta có:
t(t+2)=t2-2t+1-1=(t-1)2-1\(\ge1\)
Dấu = khi \(t=1\Rightarrow x^2+3x=1\Rightarrow\)\(x_{1,2}=\frac{-3\pm\sqrt{13}}{2}\)
Ta có: H = x(x+3)(x+1)(x+2) H = (x2+ 3x)(x2 + 3x +2) H = (x2+3x)2 + 2(x2+3x) H = (x2+3x)2 + 2(x2+3x)+1 – 1 H = (x2 + 3x +1)2 – 1 ⇔H ≥ - 1 , Dấu ‘ = ’ xảy ra khi x2 + 3x +1 = 0 ⇔x =-3+căn5 chia 2 Vậy giá trị nhỏ nhất của H là -1 khi x =-3+căn5 chia 2
\(\left|\left(x-3\right)+2\left(y-1\right)\right|\le\sqrt{\left(1+4\right)\left[\left(x-3\right)^2+\left(y-1\right)^2\right]}=5\)
\(\Rightarrow-5\le x+2y-5\le5\Rightarrow0\le x+2y\le10\)
\(P=\frac{x^2+4y^2+4xy+x+2y+9-\left(x^2-6x+9\right)-\left(y^2-2y+1\right)}{x+2y+1}\)
\(P=\frac{\left(x+2y\right)^2+\left(x+2y\right)+9-\left(x-3\right)^2-\left(y-1\right)^2}{x+2y+1}=\frac{\left(x+2y\right)^2+\left(x+2y\right)+4}{x+2y+1}\)
Đặt \(x+2y=t\ge0\)
\(P=\frac{t^2+t+4}{t+1}=t+\frac{4}{t+1}=t+1+\frac{4}{t+1}-1\)
\(P\ge2\sqrt{\frac{4\left(t+1\right)}{t+1}}-1=3\)
Ta có: P = x 2 + 2 x + 5 2 x + 1 = x + 1 2 + 4 2 x + 1 = x + 1 2 + 2 x + 1
Vì x ≥ 0 ⇒ x + 1 > 0 ⇒ x + 1 2 > 0 ; 2 x + 1 > 0
Áp dụng bất đẳng thức cô – si cho 2 số dương
x + 1 2 ; 2 x + 1 : x + 1 2 + 2 x + 1 ≥ 2 . x + 1 2 . 2 x + 1 = 2
Vậy giá trị nhỏ nhất của P là 2 khi x = 1.