a: ĐKXĐ: x>=0

Ta có: \(x-\sqrt{x}+1\)

\(=x-\sqrt{x}+\frac14+\frac34=\left(\sqrt{x}-\frac12\right)^2+\frac34\ge\frac34\forall x\) thỏa mãn ĐKXĐ

=>\(A=\frac{1}{x-\sqrt{x}+1}\le1:\frac34=\frac43\forall x\) thỏa mãn ĐKXĐ

Dấu '=' xảy ra khi \(\sqrt{x}-\frac12=0\)

=>\(\sqrt{x}=\frac12\)

=>\(x=\frac14\left(nhận\right)\)

b:

ĐKXĐ: x>=0

\(B=\frac{2x-2\sqrt{x}+5}{x-\sqrt{x}+2}\)

\(=\frac{2x-2\sqrt{x}+4}{x-\sqrt{x}+2}+\frac{1}{x-\sqrt{x}+2}=2+\frac{1}{x-\sqrt{x}+2}\)

Ta có: \(x-\sqrt{x}+2\)

\(=x-\sqrt{x}+\frac14+\frac74=\left(\sqrt{x}-\frac12\right)^2+\frac74\ge\frac74\forall x\) thỏa mãn ĐKXĐ

=>\(\frac{1}{x-\sqrt{x}+2}\le1:\frac74=\frac47\forall x\) thỏa mãn ĐKXĐ

=>\(\frac{1}{x-\sqrt{x}+2}+2\le\frac74+2=\frac{15}{4}\forall x\) thỏa mãn ĐKXĐ

Dấu '=' xảy ra khi \(\sqrt{x}-\frac12=0\)

=>\(\sqrt{x}=\frac12\)

=>\(x=\frac14\) (nhận)

Biểu thức nào em?

a.

\(y=\dfrac{3}{2}sin2x-2\left(cos^2x-sin^2x\right)+5=\dfrac{3}{2}sin2x-2cos2x+5\)

\(=\dfrac{5}{2}\left(\dfrac{3}{5}sin2x-\dfrac{4}{5}cos2x\right)+5=\dfrac{5}{2}sin\left(2x-a\right)+5\) (với \(cosa=\dfrac{3}{5}\))

\(\Rightarrow-\dfrac{5}{2}+5\le y\le\dfrac{5}{2}+5\)

b.

\(\Leftrightarrow y.sinx-2y.cosx+4y=3sinx-cosx+1\)

\(\Leftrightarrow\left(y-3\right)sinx+\left(1-2y\right)cosx=1-4y\)

Theo điều kiện có nghiệm của pt lượng giác bậc nhất:

\(\left(y-3\right)^2+\left(1-2y\right)^2\ge\left(1-4y\right)^2\)

\(\Leftrightarrow11y^2+2y-9\le0\)

\(\Leftrightarrow-1\le y\le\dfrac{9}{11}\)

c.

Do \(x^2+y^2=1\Rightarrow\) đặt \(\left\{{}\begin{matrix}x=sina\\y=cosa\end{matrix}\right.\)

\(\Rightarrow y=\dfrac{2\left(sin^2a+6sina.cosa\right)}{1+2sina.cosa+cos^2a}=\dfrac{1-cos2a+6sin2a}{1+sin2a+\dfrac{1+cos2a}{2}}=\dfrac{2-2cos2a+12sin2a}{3+2sin2a+cos2a}\)

\(\Leftrightarrow3y+2y.sin2a+y.cos2a=2-2cos2a+12sin2a\)

\(\Leftrightarrow\left(2y-12\right)sin2a+\left(y+2\right)cos2a=2-3y\)

Theo điều kiện có nghiệm của pt bậc nhất theo sin2a, cos2a:

\(\left(2y-12\right)^2+\left(y+2\right)^2\ge\left(2-3y\right)^2\)

\(\Leftrightarrow y^2+8y-36\le0\)

\(\Rightarrow-4-2\sqrt{13}\le y\le-4+2\sqrt{13}\)

a: ĐKXĐ: x>=0

Ta có: \(x-\sqrt{x}+1\)

\(=x-\sqrt{x}+\frac14+\frac34=\left(\sqrt{x}-\frac12\right)^2+\frac34\ge\frac34\forall x\) thỏa mãn ĐKXĐ

=>\(A=\frac{1}{x-\sqrt{x}+1}\le1:\frac34=\frac43\forall x\) thỏa mãn ĐKXĐ

Dấu '=' xảy ra khi \(\sqrt{x}-\frac12=0\)

=>\(\sqrt{x}=\frac12\)

=>\(x=\frac14\left(nhận\right)\)

b:

ĐKXĐ: x>=0

\(B=\frac{2x-2\sqrt{x}+5}{x-\sqrt{x}+2}\)

\(=\frac{2x-2\sqrt{x}+4}{x-\sqrt{x}+2}+\frac{1}{x-\sqrt{x}+2}=2+\frac{1}{x-\sqrt{x}+2}\)

Ta có: \(x-\sqrt{x}+2\)

\(=x-\sqrt{x}+\frac14+\frac74=\left(\sqrt{x}-\frac12\right)^2+\frac74\ge\frac74\forall x\) thỏa mãn ĐKXĐ

=>\(\frac{1}{x-\sqrt{x}+2}\le1:\frac74=\frac47\forall x\) thỏa mãn ĐKXĐ

=>\(\frac{1}{x-\sqrt{x}+2}+2\le\frac74+2=\frac{15}{4}\forall x\) thỏa mãn ĐKXĐ

Dấu '=' xảy ra khi \(\sqrt{x}-\frac12=0\)

=>\(\sqrt{x}=\frac12\)

=>\(x=\frac14\) (nhận)

\(5x^2+2xy+2y^2-\left(4x^2+4xy+y^2\right)=\left(x-y\right)^2\ge0\\ \Leftrightarrow5x^2+2xy+2y^2\ge4x^2+4xy+y^2=\left(2x+y\right)^2\)

\(\Leftrightarrow P\le\dfrac{1}{2x+y}+\dfrac{1}{2y+z}+\dfrac{1}{2z+x}=\dfrac{1}{9}\left(\dfrac{9}{x+x+y}+\dfrac{9}{y+y+z}+\dfrac{9}{z+z+x}\right)\\ \Leftrightarrow P\le\dfrac{1}{9}\left(\dfrac{1}{x}+\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{y}+\dfrac{1}{y}+\dfrac{1}{z}+\dfrac{1}{z}+\dfrac{1}{z}+\dfrac{1}{x}\right)\\ \Leftrightarrow P\le\dfrac{1}{9}\left(\dfrac{3}{x}+\dfrac{3}{y}+\dfrac{3}{z}\right)=\dfrac{1}{3}\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\right)=1\)

Dấu \("="\Leftrightarrow x=y=z=1\)

\(\sqrt{5x^2+2xy+2y^2}=\sqrt{4x^2+2xy+y^2+x^2+y^2}\ge\sqrt{4x^2+2xy+y^2+2xy}=2x+y\)

\(\Rightarrow\dfrac{1}{\sqrt{5x^2+2xy+2y^2}}\le\dfrac{1}{2x+y}=\dfrac{1}{x+x+y}\le\dfrac{1}{9}\left(\dfrac{1}{x}+\dfrac{1}{x}+\dfrac{1}{y}\right)=\dfrac{1}{9}\left(\dfrac{2}{x}+\dfrac{1}{y}\right)\)

Tương tự:

\(\dfrac{1}{\sqrt{5y^2+2yz+2z^2}}\le\dfrac{1}{9}\left(\dfrac{2}{y}+\dfrac{1}{z}\right)\) ; \(\dfrac{1}{\sqrt{5z^2+2zx+2x^2}}\le\dfrac{1}{9}\left(\dfrac{2}{z}+\dfrac{1}{x}\right)\)

Cộng vế:

\(P\le\dfrac{1}{9}\left(\dfrac{3}{x}+\dfrac{3}{y}+\dfrac{3}{z}\right)=1\)

\(P_{max}=1\) khi \(x=y=z=1\)

Câu 2:

ĐKXĐ: x<>0

\(B=\dfrac{-x^2-x-1}{x^2}\)

\(=-1-\dfrac{1}{x}-\dfrac{1}{x^2}\)

\(=-\left(\dfrac{1}{x^2}+\dfrac{1}{x}+1\right)\)

\(=-\left(\dfrac{1}{x^2}+2\cdot\dfrac{1}{x}\cdot\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{3}{4}\right)\)

\(=-\left(\dfrac{1}{x}+\dfrac{1}{2}\right)^2-\dfrac{3}{4}< =-\dfrac{3}{4}\forall x< >0\)

Dấu '=' xảy ra khi 1/x+1/2=0

=>1/x=-1/2

=>x=-2

\(y=x+\dfrac{1}{x}-5\ge2\sqrt{\dfrac{x}{x}}-5=-3\)

\(y_{min}=-3\) khi \(x=1\)

\(y=4x^2+\dfrac{1}{2x}+\dfrac{1}{2x}-4\ge3\sqrt[3]{\dfrac{4x^2}{2x.2x}}-4=-1\)

\(y_{min}=-1\) khi \(x=\dfrac{1}{2}\)

\(y=x+\dfrac{4}{x}\Rightarrow y'=1-\dfrac{4}{x^2}=0\Rightarrow x=-2\)

\(y\left(-2\right)=-4\Rightarrow\max\limits_{x>0}y=-4\) khi \(x=-2\)

\(a,B=4,2+\left|x+1,5\right|\ge4,2\\ B_{min}=4,2\Leftrightarrow x+1,5=0\Leftrightarrow x=-1,5\\ b,C=\dfrac{4}{5}-\left|2x+1\right|\le\dfrac{4}{5}\\ C_{max}=\dfrac{4}{5}\Leftrightarrow2x+1=0\Leftrightarrow x=-\dfrac{1}{2}\)

a, Do |x +1,5| ≥ 0 ⇒ 4,2 + |x + 1,5| ≥ 4,2

Dấu "=" xảy ra ⇔ x + 1,5 = 0 ⇔  x = - 1,5

Vậy B_min=  4,2 ⇔ x= -1,5

b, Do |2x + 1| ≥ 0 ⇒ \(\dfrac{4}{5}-\left|2x+1\right|\le\dfrac{4}{5}\)

Dấu "=" xảy ra ⇔ 2x + 1 = 0 ⇔ 2x = -1 ⇔ \(x=-\dfrac{1}{2}\)

Vậy C_max = \(\dfrac{4}{5}\Leftrightarrow x=-\dfrac{1}{2}\)