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\(\left(sin\dfrac{x}{2}-cox\dfrac{x}{2}\right)^2+\sqrt{3}cosx=2sin5x+1\)

⇔\(sin^2\dfrac{x}{2}+cos^2\dfrac{x}{2}-2sin\dfrac{x}{2}cos\dfrac{x}{2}+\sqrt{3}cosx=2sin5x+1\)

⇔\(1-sinx+\sqrt{3}cosx=2sin5x+1\)

⇔\(sin\left(\dfrac{\Pi}{3}-x\right)=sin5x\)

\(2sinx\left(\sqrt{3}cosx+sinx+2sin3x\right)=1\)

⇔\(2\sqrt{3}sinxcosx+2sin^2x+4sinxsin3x=1\)

⇔\(\sqrt{3}sin2x+1-cos2x+cos2x-2cos4x=1\)

⇔\(\sqrt{3}sin2x+cos2x=2cos4x\)

⇔\(cos\left(2x-\dfrac{\Pi}{3}\right)=cos4x\)

C

a/ ĐKXĐ:

\(1-sinx>0\Leftrightarrow sinx\ne1\)

\(\Rightarrow x\ne\frac{\pi}{2}+k2\pi\)

b/ ĐKXĐ:

\(\left\{{}\begin{matrix}sinx.cosx\ne0\\tanx+4cotx+2\ne0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}sin2x\ne0\\tan^2x+2tanx+4\ne0\left(luôn-đúng\right)\end{matrix}\right.\)

\(\Rightarrow2x\ne k\pi\Rightarrow x\ne\frac{k\pi}{2}\)

c/

\(\left\{{}\begin{matrix}\frac{1-cosx}{2+cosx}\ge0\\2+cosx\ne0\end{matrix}\right.\) \(\Rightarrow x\in R\)

câu 1 ntn.

gọi số thú săn đc mỗi ng là a1, a2,..., a7

vì mỗi người ăn đc số thú khác nhau nên giả sử là a1<a2<ả3<...<a7

TH1: $a5>15\Rightarrow a5+a6+a7\ge 16+17+18=51>50$

TH2 : $a5\le 15\Rightarrow a1+a2+a3+a4\le 14+13+12+11=50\Rightarrow$$a5+a6+a7\ge 50$


câu 2.

Xét $F\left(x\right)=a_{0}x+a_1.sinx+a_2.\frac{sin2x}{2}+...+a_n.\frac{sinnx}{n}$

Nhìn thấy đạo hàm bằng định nghĩa là thấy ớn, dài dữ dội

- Khi \(x>1\) \(\Rightarrow f\left(x\right)=\frac{4x-4}{x+1}\)

\(\Delta x=x-x_0\) \(\Rightarrow\Delta y=\frac{4\Delta x+4x_0-4}{x_0+\Delta x+1}-\frac{4x_0-4}{x_0+1}=\frac{8\Delta x}{\left(x_0+1\right)\left(x_0+1+\Delta x\right)}\)

\(\Rightarrow f'\left(x_0\right)=\lim\limits_{\Delta x\rightarrow0}\frac{8\Delta x}{\Delta x\left(x_0+1\right)\left(x_0+1+\Delta x\right)}=\frac{8}{\left(x_0+1\right)^2}\)

- Khi \(x< 1\Rightarrow f\left(x\right)=2x-2\)

\(\Delta x\) là số gia của \(x_0< 1\)

\(\Rightarrow\Delta y=2\left(x_0+\Delta x\right)-2-\left(2x_0-2\right)=2\Delta x\)

\(\Rightarrow f'\left(x_0\right)=\lim\limits_{\Delta x\rightarrow0}\frac{2\Delta x}{\Delta x}=2\)

- Khi \(x\rightarrow1^+\Rightarrow\Delta y\rightarrow2\left(1+\Delta x\right)-2\rightarrow2\Delta x\)

\(\lim\limits_{x\rightarrow1^+}f'\left(x\right)=\lim\limits_{\Delta x\rightarrow0}\frac{2\Delta x}{\Delta x}=2\)

\(\lim\limits_{x\rightarrow1^-}f'\left(x\right)=\lim\limits_{x\rightarrow1^-}\frac{8}{\left(1+1\right)^2}=2\)

\(\Rightarrow f'\left(1\right)=2\)

a/ ĐKXĐ: ...

\(y'=\frac{1}{2\sqrt{x+3}}-\frac{1}{2\sqrt{1-x}}\)

\(y'=0\Leftrightarrow\frac{1}{2\sqrt{x+3}}=\frac{1}{2\sqrt{1-x}}\Leftrightarrow\sqrt{x+3}=\sqrt{1-x}\)

\(\Leftrightarrow x+3=1-x\Rightarrow x=-1\)

b/ \(y'=\frac{sinx-x.cosx}{sin^2x}\)

c/ \(y'=\frac{cosx}{2\sqrt{x}}-\sqrt{x}.sinx\)

a/ \(y=3x+2\)

b/ \(y=-\frac{1}{4}x+1\)

c/ \(y=\frac{1}{6}x+\frac{3}{2}\)

d/ \(y=-32x-48\)