\(y'=\dfrac{\left(-2x+2\right)\left(x-3\right)-\left(-x^2+2x+c\right)}{\left(x-3\right)^2}=\dfrac{-x^2+6x-6-c}{\left(x-3\right)^2}\)

\(\Rightarrow\) Cực đại và cực tiểu của hàm là nghiệm của: \(-x^2+6x-6-c=0\) (1)

\(\Delta'=9-\left(6+c\right)>0\Rightarrow c< 3\)

Gọi \(x_1;x_2\) là 2 nghiệm của (1) \(\Rightarrow\left\{{}\begin{matrix}-x_1^2+6x_1-6=c\\-x_2^2+6x_2-6=c\end{matrix}\right.\)

\(\Rightarrow m-M=\dfrac{-x_1^2+2x_1+c}{x_1-3}-\dfrac{-x_2^2+2x_2+c}{x_2-3}=4\)

\(\Leftrightarrow\dfrac{-2x_1^2+8x_1-6}{x_1-3}-\dfrac{-2x_2^2+8x_2-6}{x_2-3}=4\)

\(\Leftrightarrow2\left(1-x_1\right)-2\left(1-x_2\right)=4\)

\(\Leftrightarrow x_2-x_1=2\)

Kết hợp với Viet: \(\left\{{}\begin{matrix}x_2-x_1=2\\x_1+x_2=6\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x_1=2\\x_2=4\end{matrix}\right.\)

\(\Rightarrow c=2\)

Có 1 giá trị nguyên

2.

\(\Leftrightarrow cos2x-cos8x-sin3x+cos5x-2sin5x.cos5x=0\)

\(\Leftrightarrow2sin5x.sin3x-sin3x+cos5x-2sin5x.cos5x=0\)

\(\Leftrightarrow sin3x\left(2sin5x-1\right)-cos5x\left(2sin5x-1\right)=0\)

\(\Leftrightarrow\left(sin3x-cos5x\right)\left(2sin5x-1\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}cos5x=sin3x=cos\left(\dfrac{\pi}{2}-3x\right)\\sin5x=\dfrac{1}{2}\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}5x=\dfrac{\pi}{2}-3x+k2\pi\\5x=3x-\dfrac{\pi}{2}+k2\pi\\5x=\dfrac{\pi}{6}+k2\pi\\5x=\dfrac{5\pi}{6}+k2\pi\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{16}+\dfrac{k\pi}{4}\\x=-\dfrac{\pi}{4}+k\pi\\x=\dfrac{\pi}{30}+\dfrac{k2\pi}{5}\\x=\dfrac{\pi}{6}+\dfrac{k2\pi}{5}\end{matrix}\right.\)

3.

\(\Leftrightarrow1+sinx=cosx-cos3x+2sinx.cosx+1-2sin^2x\)

\(\Leftrightarrow sinx=2sin2x.sinx+2sinx.cosx-2sin^2x\)

\(\Leftrightarrow\left[{}\begin{matrix}sinx=0\Rightarrow x=k\pi\\1=2sin2x+2cosx-2sinx\left(1\right)\end{matrix}\right.\)

\(\left(1\right)\Leftrightarrow4sinx.cosx+2cosx-2sinx-1=0\)

\(\Leftrightarrow2cosx\left(2sinx+1\right)-\left(2sinx+1\right)=0\)

\(\Leftrightarrow\left(2cosx+1\right)\left(2sinx-1\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}sinx=\dfrac{1}{2}\\cosx=-\dfrac{1}{2}\end{matrix}\right.\)

\(\Leftrightarrow...\)

is not different

is not different 

1)

\((x+2)(x+3)(x+4)(x+5)-24\\=[(x+2)(x+5)]\cdot[(x+3)(x+4)]-24\\=(x^2+7x+10)(x^2+7x+12)-24\)

Đặt \(x^2+7x+10=y\), khi đó biểu thức trở thành:

\(y(y+2)-24\\=y^2+2y-24\\=y^2+2y+1-25\\=(y+1)^2-5^2\\=(y+1-5)(y+1+5)\\=(y-4)(y+6)\\=(x^2+7x+10-4)(x^2+7x+10+6)\\=(x^2+7x+6)(x^2+7x+16)\)

2) Bạn xem lại đề!

1/ \(y'=\left(1-3x\right)'\sqrt{x-3}+\left(1-3x\right)\left(\sqrt{x-3}\right)'=-3\sqrt{x-3}+\dfrac{1}{2\sqrt{x-3}}\left(1-3x\right)\)

2/ \(y'=\dfrac{1}{\sqrt{2x+1}}-\dfrac{1}{\left(x+1\right)^2}\)

3/ \(y'=\dfrac{1}{2}.\sqrt{\dfrac{1+x}{1-x}}.\left(\dfrac{1-x}{1+x}\right)'=\dfrac{1}{2}\sqrt{\dfrac{1+x}{1-x}}.\dfrac{-2}{\left(1+x\right)^2}=-\sqrt{\dfrac{1+x}{1-x}}.\dfrac{1}{\left(1+x\right)^2}\)

4/ \(y'=\left(\cos5x\right)'.\cos7x+\cos5x.\left(\cos7x\right)'=-5\sin5x.\cos7x-7\cos5x\sin7x\)

5/ \(y'=\left(\cos x\right)'\sin^2x+\cos x\left(\sin^2x\right)'=-\sin^3x+2\sin x.\cos^2x\)

6/ \(y'=\left(\tan^42x\right)'=4.\tan^32x.\dfrac{2}{\cos^22x}\)

7/ \(y'=\dfrac{2\sin x+2\cos x-2x.\cos x+2x\sin x}{\left(\sin x+\cos x\right)^2}\)

Ờm, bạn tự rút gọn nhé :) Mình đang hơi lười :b

Lời giải:
$A=\cos 2x-2\sin 5x\sin x=\cos 2x-2.\frac{-1}{2}[\cos (5x+x)-\cos (5x-x)]$

$=\cos 2x+\cos 6x-\cos 4x$

$=(\cos 2x+\cos 6x)-\cos 4x$

$=2\cos \frac{2x+6x}{2}\cos \frac{6x-2x}{2}-\cos 4x$

$=2\cos 4x\cos 2x-\cos 4x$

$=\cos 4x[2\cos 2x-1]$

Những đáp án A,B,C,D bạn đưa ra không có đáp án nào đúng cả.