Ta có : \(\left\{{}\begin{matrix}a=1>0\\\Delta'=\left(m-1\right)^2-\left(m-2\right)=m^2-3m+3>0\end{matrix}\right.\)

Để \(f\left(x\right)\le0\forall x\in\left[0;1\right]\)

\(\Leftrightarrow x_1\le0< 1\le x_2\)

\(\Rightarrow\left\{{}\begin{matrix}f\left(0\right)\le0\\f\left(1\right)\le0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}m-2\le0\\-m+1\le0\end{matrix}\right.\Rightarrow1\le m\le2\)

Vậy ...

\(f\left(x\right)=x^2-2\left(m-1\right)x+m-2\)

Yêu cầu bài toán thõa mãn khi \(f\left(x\right)=0\) có hai nghiệm thỏa mãn \(x_1\le1< 3\le x_2\)

\(\left\{{}\begin{matrix}\Delta'=m^2-3m+3\ge0\\f\left(1\right)\le0\\f\left(3\right)\le0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}m\in R\\-m+1\le0\\15-5m\le0\end{matrix}\right.\)

\(\Leftrightarrow m\ge3\)

\(a=1>0\) ; \(\Delta'=\left(m-2\right)^2-\left(m-2\right)=\left(m-2\right)\left(m-3\right)\)

a/ Để \(f\left(x\right)\le0\) \(\forall x\in\left(0;1\right)\)

\(\Leftrightarrow x_1\le0< 1\le x_2\)

\(\Leftrightarrow\left\{{}\begin{matrix}f\left(0\right)\le0\\f\left(1\right)\le0\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}m-2\le0\\1-\left(m-2\right)\le0\end{matrix}\right.\) \(\Rightarrow\) ko tồn tại m thỏa mãn

Do đó các câu c, f cũng không tồn tại m thỏa mãn

b/ TH1: \(\Delta< 0\Rightarrow2< m< 3\)

TH2: \(\left\{{}\begin{matrix}\Delta=0\\-\frac{b}{2a}\notin\left(0;1\right)\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}m=2\\m=3\end{matrix}\right.\)

TH3: \(\left\{{}\begin{matrix}\Delta>0\\\left[{}\begin{matrix}0\le x_1< x_2\\x_1< x_2\le1\end{matrix}\right.\end{matrix}\right.\)

\(\Delta>0\Rightarrow\left[{}\begin{matrix}m>3\\m< 2\end{matrix}\right.\)

\(0\le x_1< x_2\Leftrightarrow\left\{{}\begin{matrix}f\left(0\right)\ge0\\\frac{x_1+x_2}{2}>0\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}m-2\ge0\\m-2>0\end{matrix}\right.\) \(\Rightarrow m>2\) \(\Rightarrow m>3\)

\(x_1< x_2\le1\Leftrightarrow\left\{{}\begin{matrix}f\left(1\right)\ge0\\\frac{x_1+x_2}{2}< 1\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}3-m\ge0\\m-2< 1\end{matrix}\right.\) \(\Rightarrow\) ko tồn tại m

Kết hợp 3 TH \(\Rightarrow m\ge2\)

d/ Tương tự như câu b, nhưng

TH2: \(\left\{{}\begin{matrix}\Delta=0\\-\frac{b}{2a}\in\left[0;1\right]\end{matrix}\right.\) \(\Rightarrow\) không tồn tại m thỏa mãn

TH3: \(\left\{{}\begin{matrix}\Delta>0\\\left[{}\begin{matrix}0< x_1< x_2\\x_1< x_2< 1\end{matrix}\right.\end{matrix}\right.\)

\(\Rightarrow m>3\)

Kết hợp 3 TH \(\Rightarrow\left[{}\begin{matrix}2< m< 3\\m>3\end{matrix}\right.\)

e/

TH1: \(\Delta\le0\Rightarrow2\le m\le3\)

TH2: \(\left\{{}\begin{matrix}\Delta>0\\\left[{}\begin{matrix}0\le x_1< x_2\\x_1< x_2\le1\end{matrix}\right.\end{matrix}\right.\) \(\Rightarrow m>3\)

\(\Rightarrow m\ge2\)

1.a.

\(\left(x+1\right)\left(x+2\right)\left(x-2\right)\left(x+5\right)\ge m\)

\(\Leftrightarrow\left(x^2+3x+2\right)\left(x^2+3x-10\right)\ge m\)

Đặt \(x^2+3x-10=t\ge-\dfrac{49}{4}\)

\(\Rightarrow\left(t+2\right)t\ge m\Leftrightarrow t^2+2t\ge m\)

Xét \(f\left(t\right)=t^2+2t\) với \(t\ge-\dfrac{49}{4}\)

\(-\dfrac{b}{2a}=-1\) ; \(f\left(-1\right)=-1\) ; \(f\left(-\dfrac{49}{4}\right)=\dfrac{2009}{16}\)

\(\Rightarrow f\left(t\right)\ge-1\)

\(\Rightarrow\) BPT đúng với mọi x khi \(m\le-1\)

Có 30 giá trị nguyên của m

1b.

Với \(x=0\)  BPT luôn đúng

Với \(x\ne0\) BPT tương đương:

\(\dfrac{\left(x^2-2x+4\right)\left(x^2+3x+4\right)}{x^2}\ge m\)

\(\Leftrightarrow\left(x+\dfrac{4}{x}-2\right)\left(x+\dfrac{4}{x}+3\right)\ge m\)

Đặt \(x+\dfrac{4}{x}-2=t\) \(\Rightarrow\left[{}\begin{matrix}t\ge2\\t\le-6\end{matrix}\right.\)

\(\Rightarrow t\left(t+5\right)\ge m\Leftrightarrow t^2+5t\ge m\)

Xét hàm \(f\left(t\right)=t^2+5t\) trên \(D=(-\infty;-6]\cup[2;+\infty)\)

\(-\dfrac{b}{2a}=-\dfrac{5}{2}\notin D\) ; \(f\left(-6\right)=6\) ; \(f\left(2\right)=14\)

\(\Rightarrow f\left(t\right)\ge6\)

\(\Rightarrow m\le6\)

Vậy có 37 giá trị nguyên của m thỏa mãn

2.

Xét với \(x\ge1\)

\(m\left(x+1\right)+3\left(x-1\right)-2\sqrt{x^2-1}=0\)

\(\Leftrightarrow m+3\left(\dfrac{x-1}{x+1}\right)-2\sqrt{\dfrac{x-1}{x+1}}=0\)

Đặt \(\sqrt{\dfrac{x-1}{x+1}}=t\Rightarrow0\le t< 1\)

\(\Rightarrow m+3t^2-2t=0\)

\(\Leftrightarrow3t^2-2t=-m\)

Xét hàm \(f\left(t\right)=3t^2-2t\) trên \(D=[0;1)\)

\(-\dfrac{b}{2a}=\dfrac{1}{3}\in D\) ; \(f\left(0\right)=0\) ; \(f\left(\dfrac{1}{3}\right)=-\dfrac{1}{3}\) ; \(f\left(1\right)=1\)

\(\Rightarrow-\dfrac{1}{3}\le f\left(t\right)< 1\)

\(\Rightarrow\) Pt có nghiệm khi \(-\dfrac{1}{3}\le-m< 1\)

\(\Leftrightarrow-1< m\le\dfrac{1}{3}\)

Câu 1:

Đặt \(3^x=t(t>0)\)

PT trở thành:

\(t^2-6.t+5=m\)

\(\Leftrightarrow t^2-6t+(5-m)=0\)

Để PT có đúng một nghiệm thì \(\Delta'=9-(5-m)=0\)

\(\Leftrightarrow m=-4\)

Thử lại \(9^x-6.3^x+9=0\Leftrightarrow 3^x=3\Leftrightarrow x=1\in [0;+\infty )\) (đúng)

Vậy \(m=-4\)

Câu 2:

\(4^x-2^x-m\geq 0\Leftrightarrow (2^x)^2-2^x-m\geq 0\)

Đặt \(2^x=t\Rightarrow t^2-t-m\geq 0\) với mọi \(t\in (1; 2)\)

\(\Leftrightarrow m\leq t^2-t\Leftrightarrow m\leq \min (t^2-t)\)

Xét hàm \(f(t)=t^2-t\Rightarrow f'(t)=2t-1>0\forall t\in (1;2)\)

\(\Rightarrow f(t)> f(1)=0\) với mọi \(t\in (1;2)\)

Do đó \(m\leq 0\)

Câu 3:

Đặt \(2^x=t\Rightarrow t\in \left(\frac{1}{2}; 4\right)\)

BPT \(\Leftrightarrow t^2-4t-m\leq 0\Leftrightarrow m\geq t^2-4t\)

Để HPT luôn đúng với x thuộc khoảng xác định thì \(m\geq \max (t^2-4t)\)

Xét \(f(t)=t^2-4t\Rightarrow f'(t)=2t-4=0\Leftrightarrow t=2\)

Lập bảng biến thiên suy ra \(f(t)< f(4)=0\)

Do đó \(m\geq 0\)

\(ycđb\Leftrightarrow\left[{}\begin{matrix}\Delta\le0\Leftrightarrow\left(m-5\right)^2\le0\Leftrightarrow m=5\\1\le x1< x2\left(1\right)\\x1< x2\le-1\left(2\right)\end{matrix}\right.\)

\(\Delta>0\Leftrightarrow\left(m-5\right)^2>0\Leftrightarrow m\ne5\)

\(\left(1\right)\Leftrightarrow\left\{{}\begin{matrix}\left(x1-1\right)\left(x2-1\right)\ge0\\x1+x2-2>0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x1x2-\left(x1+x2\right)+1\ge0\\5m-5-2>0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}6m^2-10m-\left(5m-5\right)+1\ge0\\m>\dfrac{7}{5}\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\left[{}\begin{matrix}m\le\dfrac{1}{2}\\m\ge2\end{matrix}\right.\\m>\dfrac{7}{5}\end{matrix}\right.\)

\(\Rightarrow\left\{{}\begin{matrix}m\ge2\\m\ne5\end{matrix}\right.\)

\(\left(2\right)\Leftrightarrow\left\{{}\begin{matrix}\left(x1+1\right)\left(x2+1\right)\ge0\\x1+x2+2< 0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x1x2+x1+x2+1\ge0\\x1+x2+2< 0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}6m^2-10m+5m-5+1\ge0\\5m-5-2< 0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\left[{}\begin{matrix}m\le-\dfrac{1}{2}\\m\ge\dfrac{4}{3}\end{matrix}\right.\\m< \dfrac{7}{5}\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}m\in[\dfrac{4}{3};\dfrac{7}{5})\\m\le-\dfrac{1}{2}\end{matrix}\right.\)

\(\Rightarrow\)\(m\in(-\infty;-\dfrac{1}{2}]\cup[\dfrac{4}{3};\dfrac{7}{5})\cup[2;+\infty)\cup\left\{5\right\}\)

\(\text{f(x)}\)\(\text{>0}\)\(\text{⇔}\)\(\text{2x}\)²\(\text{-3x+1}\)\(>0\)⇔\(\left\{{}\begin{matrix}x>1\\x< \dfrac{1}{2}\end{matrix}\right.\)

⇒x∈(−∞;\(\dfrac{1}{2}\))∪(1;+∞)