Ta có: \(A=-2x^2-5x+3\)

\(=-2\left(x^2+\dfrac{5}{2}x-\dfrac{3}{2}\right)\)

\(=-2\left(x^2+2\cdot x\cdot\dfrac{5}{4}+\dfrac{25}{16}-\dfrac{49}{16}\right)\)

\(=-2\left(x+\dfrac{5}{4}\right)^2+\dfrac{49}{8}\)

Ta có: \(\left(x+\dfrac{5}{4}\right)^2\ge0\forall x\)

\(\Rightarrow-2\left(x+\dfrac{5}{4}\right)^2\le0\forall x\)

\(\Rightarrow-2\left(x+\dfrac{5}{4}\right)^2+\dfrac{49}{8}\le\dfrac{49}{8}\forall x\)

Dấu '=' xảy ra khi \(x+\dfrac{5}{4}=0\)

hay \(x=-\dfrac{5}{4}\)

Vậy: Giá trị lớn nhất của biểu thức \(A=-2x^2-5x+3\) là \(\dfrac{49}{8}\) khi \(x=-\dfrac{5}{4}\)

Lời giải:

a) 

Áp dụng BĐT Bunhiacopxky:

\((y-2x)^2\leq (16y^2+36x^2)(\frac{1}{16}+\frac{1}{9})=9.\frac{25}{144}\)

\(\Rightarrow \frac{-5}{4}\leq y-2x\leq \frac{5}{4}\Rightarrow \frac{15}{4}\leq y-2x+5\leq \frac{25}{4}\)

Vậy $A_{\min}=\frac{15}{4}$ và $A_{\max}=\frac{25}{4}$

b) 

Áp dụng BĐT Bunhiacopxky:

\((2x-y)^2\leq (\frac{x^2}{4}+\frac{y^2}{9})(16+9)=25\)

\(\Rightarrow -5\leq 2x-y\leq 5\Leftrightarrow -7\leq 2x-y-2\leq 3\)

Vậy $B_{min}=-7; B_{\max}=3$

\(a,\) Đặt \(A=\dfrac{3x^2-2x+3}{x^2+1}\Leftrightarrow Ax^2+A=3x^2-2x+3\)

\(\Leftrightarrow x^2\left(A-3\right)-2x+A-3=0\)

Coi đây là PT bậc 2 ẩn x, PT có nghiệm 

\(\Leftrightarrow\Delta=4-4\left(A-3\right)^2\ge0\\ \Leftrightarrow\left(A-3\right)^2\le1\Leftrightarrow2\le A\le4\)

Vậy \(A_{min}=4\Leftrightarrow\dfrac{3x^2-2x+3}{x^2+1}=4\Leftrightarrow x=...\)

\(b,\) Đặt \(B=\dfrac{3x^2-4x+4}{x^2+2}\Leftrightarrow Bx^2+2B=3x^2-4x+4\)

\(\Leftrightarrow x^2\left(B-3\right)+4x+2B-4=0\)

Coi đây là PT bậc 2 ẩn x, PT có nghiệm

\(\Leftrightarrow\Delta=16-8\left(B-2\right)\left(B-3\right)\ge0\\ \Leftrightarrow\left(B-2\right)\left(B-3\right)\le2\\ \Leftrightarrow B^2-5B+4\le0\\ \Leftrightarrow\left(B-1\right)\left(B-4\right)\le0\\ \Leftrightarrow1\le B\le4\)

Vậy\(B_{min}=4\Leftrightarrow\dfrac{3x^2-4x+4}{x^2+2}=4\Leftrightarrow x=...\)

a, \(y=sin^2x-2sinx+3cos^2x\)

\(=sin^2x-2sinx+3\left(1-sin^2x\right)\)

\(=3-2sinx-2sin^2x\)

Đặt \(sinx=t\left(t\in\left[0;1\right]\right)\)

\(\Rightarrow y=f\left(t\right)=3-2t-2t^2\)

\(\Rightarrow y_{min}=min\left\{f\left(0\right);f\left(1\right)\right\}=-1\)

\(y_{max}=max\left\{f\left(0\right);f\left(1\right)\right\}=3\)

b, \(y=sinx-cosx+sin2x+5\)

\(=sinx-cosx-\left(sinx-cosx\right)^2+6\)

Đặt \(sinx-cosx=t\left(t\in\left[-\sqrt{2};\sqrt{2}\right]\right)\)

\(\Rightarrow y=f\left(t\right)=-t^2+t+6\)

\(\Rightarrow y_{min}=min\left\{f\left(-\sqrt{2}\right);f\left(0\right)\right\}=4-\sqrt{2}\)

\(y_{max}=max\left\{f\left(-\sqrt{2}\right);f\left(0\right)\right\}=6\)

c, \(y=sinx-cosx+sinx.cosx-3\)

\(=sinx-cosx-\dfrac{1}{2}\left(sinx-cosx\right)^2-\dfrac{5}{2}\)

Đặt \(sinx-cosx=t\left(t\in\left[-\sqrt{2};\sqrt{2}\right]\right)\)

\(\Rightarrow y=f\left(t\right)=-\dfrac{1}{2}t^2+t-\dfrac{5}{2}\)

\(\Rightarrow y_{min}=min\left\{f\left(-\sqrt{2}\right);f\left(\sqrt{2}\right);f\left(1\right)\right\}=-\dfrac{7+2\sqrt{2}}{2}\)

\(y_{max}=max\left\{f\left(-\sqrt{2}\right);f\left(\sqrt{2}\right);f\left(1\right)\right\}=-2\)

\(\left|x-\dfrac{2}{3}\right|-4\ge-4\forall x\)

Dấu '=' xảy ra khi \(x=\dfrac{2}{3}\)

\(H=\left(3x-6\right)^2-3\left|2x-4\right|+2023\)

\(=\left(3x-6\right)^2-2\left|3x-6\right|+2023\)

\(=\left(3x-6\right)^2-2\left|3x-6\right|+1+2022\)

\(=\left(\left|3x-6\right|-1\right)^2+2022\)

Do \(\left(\left|3x-6\right|-1\right)^2\ge0;\forall x\)

\(\Rightarrow H\ge2022\)

\(\Rightarrow H_{min}=2022\) khi \(\left|3x-6\right|-1=0\Rightarrow x=\left\{\dfrac{7}{3};\dfrac{5}{3}\right\}\)

\(P=5x^2+y^2-2x(y+8)+2023\\=5x^2+y^2-2xy-16x+2023\\=(x^2-2xy+y^2)+(4x^2-16x+16)+2007\\=(x-y)^2+4(x^2-4x+4)+2007\\=(x-y)^2+4(x-2)^2+2007\)

Ta thấy: \(\left(x-y\right)^2\ge0\forall x;y\)

              \(4\left(x-2\right)^2\ge0\forall x\)

\(\Rightarrow (x-y)^2+4(x-2)^2\ge0\forall x;y\\\Rightarrow P=(x-y)^2+4(x-2)^2+2007\ge2007\forall x;y\)

Dấu \("="\) xảy ra khi: \(\left\{{}\begin{matrix}x-y=0\\x-2=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=y\\x=2\end{matrix}\right.\Leftrightarrow x=y=2\)

Vậy \(Min_P=2007\) khi \(x=y=2\).

\(\text{#}Toru\)

\(P=5x^2+y^2-2x\left(y+8\right)+2023\)

\(=x^2-2xy+y^2+4x^2-16x+2023\)

\(=\left(x-y\right)^2+4x^2-16x+16+2007\)

\(=\left(x-y\right)^2+\left(2x-4\right)^2+2007>=2007\)

Dấu = xảy ra khi x-y=0 và 2x-4=0

=>x=y=2

P = 5x² + y² - 2x(y + 8) + 2023

= 4x² + x² + y² - 2xy - 16x + 16 + 2007

= (x² - 2xy + y²) + (4x² - 16x + 16) + 2007

= (x - y)² + (2x - 4)² + 2007

Do (x - y)² ≥ 0 với mọi x, y R

(2x - 4)² ≥ 0 với mọi x R

⇒ (x - y)² + (2x - 4)² ≥ 0 với mọi x, y ∈ R

⇒ (x - y)² + (2x - 4)² + 2007 ≥ 2007 với mọi x, y ∈ R

Vậy GTNN của P là 2007 khi x = y = 2

b) Ta có: \(\left|x+4\right|\ge0\forall x\)

\(\Leftrightarrow\left|x+4\right|+1996\ge1996\forall x\)

Dấu '=' xảy ra khi x=-4