a)\(y=\sqrt{3}sinx+cosx=2\left(\dfrac{\sqrt{3}}{2}sinx+\dfrac{1}{2}cosx\right)\)\(=2\left(sinx.cos\dfrac{\pi}{6}+cosx.sin\dfrac{\pi}{6}\right)\)\(=2sin\left(x+\dfrac{\pi}{6}\right)\)

Có \(-1\le sin\left(x+\dfrac{\pi}{6}\right)\le1\) \(\Leftrightarrow-2\le2sin\left(x+\dfrac{\pi}{6}\right)\le2\)

\(\Leftrightarrow-2\le y\le2\)

miny=-2 \(\Leftrightarrow sin\left(x+\dfrac{\pi}{6}\right)=-1\)  \(\Leftrightarrow x+\dfrac{\pi}{6}=-\dfrac{\pi}{2}+2k\pi\left(k\in Z\right)\) \(\Leftrightarrow x=-\dfrac{2\pi}{3}+k2\pi\left(k\in Z\right)\)

maxy=2\(\Leftrightarrow sin\left(x+\dfrac{\pi}{6}\right)=1\) \(\Leftrightarrow x+\dfrac{\pi}{6}=\dfrac{\pi}{2}+k2\pi\left(k\in Z\right)\)\(\Leftrightarrow x=\dfrac{\pi}{3}+k2\pi\left(k\in Z\right)\)

b) \(y=sin2x-cos2x=\sqrt{2}sin\left(2x-\dfrac{\pi}{4}\right)\)

Có \(\sqrt{2}\ge\sqrt{2}sin\left(2x-\dfrac{\pi}{4}\right)\ge-\sqrt{2}\)

\(\Leftrightarrow\sqrt{2}\ge y\ge-\sqrt{2}\)

miny=\(-\sqrt{2}\) \(\Leftrightarrow sin\left(2x-\dfrac{\pi}{4}\right)=-1\)\(\Leftrightarrow2x-\dfrac{\pi}{4}=-\dfrac{\pi}{2}+k2\pi\left(k\in Z\right)\)\(\Leftrightarrow x=-\dfrac{\pi}{8}+k\pi\left(k\in Z\right)\)

maxy=\(\sqrt{2}\Leftrightarrow sin\left(2x-\dfrac{\pi}{4}\right)=1\)\(\Leftrightarrow x=\dfrac{3\pi}{8}+k\pi\left(k\in Z\right)\)

c) \(y=3sinx+4cosx=5\left(\dfrac{3}{5}sinx+\dfrac{4}{5}cosx\right)\)

Đặt \(cosa=\dfrac{3}{5}\) và \(sina=\dfrac{4}{5}\)(vì cos²a+sin²a=1)

\(y=5\left(sinx.cosa+cosx.sina\right)\)\(=5sin\left(x+a\right)\)

\(\Rightarrow-5\le y\le5\)

miny=-5 <=> \(sin\left(x+a\right)=-1\)\(\Leftrightarrow x=-\dfrac{\pi}{2}-arc.sina+k2\pi\left(k\in Z\right)\)

maxy=5 <=> \(sin\left(x+a\right)=1\)\(\Leftrightarrow x=\dfrac{\pi}{2}-arc.sina+k2\pi\left(k\in Z\right)\)

(P/s1:cái x ở câu c ấy trông nó ngu ngu??
 P/s2:sau khi load lại câu hỏi ở 1 tab khác ,thấy 1 câu trả lời nhưng vẫn đăng vì cảm thấy bỏ đi hơi phí :?)

Áp dụng quy tắc sau: Nếu \(a\sin x+b\cos y=c\Leftrightarrow a^2+b^2\ge c^2\)

a/ \(3+1\ge y^2\Leftrightarrow4\ge y^2\Leftrightarrow-2\le y\le2\)

\(y_{max}=2\Leftrightarrow\sqrt{3}\sin x+\cos x=2\Leftrightarrow\dfrac{\sqrt{3}}{2}\sin x+\dfrac{1}{2}\cos x=1\Leftrightarrow\cos\dfrac{\pi}{6}.\sin x+\sin\dfrac{\pi}{6}.\cos x=1\)

\(\Rightarrow\sin\left(x+\dfrac{\pi}{6}\right)=1\Leftrightarrow x+\dfrac{\pi}{6}=\dfrac{\pi}{2}+k2\pi\Leftrightarrow x=\dfrac{\pi}{3}+k2\pi\)

\(y_{min}=-2\Leftrightarrow\sin\left(x+\dfrac{\pi}{6}\right)=-1\Leftrightarrow x+\dfrac{\pi}{6}=-\dfrac{\pi}{2}+k2\pi\Leftrightarrow x=-\dfrac{2}{3}\pi+k2\pi\)

ĐKXĐ: \(sinx;cosx\ge0\)

Do \(\left\{{}\begin{matrix}0\le sinx\le1\\0\le cosx\le1\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}\sqrt{sinx}\ge sin^2x\\\sqrt{cosx}\ge cos^2x\end{matrix}\right.\)

\(\Rightarrow\sqrt{sinx}+\sqrt{cosx}\ge sin^2x+cos^2x=1\)

\(\Rightarrow y_{min}=1\) (khi \(x=\dfrac{\pi}{2}+k2\pi\) hoặc \(k2\pi\))

Mặt khác áp dụng Bunhiacopxki:

\(y\le\sqrt{2\left(sinx+cosx\right)}\le\sqrt{2\sqrt{2\left(sin^2x+cos^2x\right)}}=\sqrt[4]{8}\)

\(y_{max}=\sqrt[4]{8}\) khi \(x=\dfrac{\pi}{4}+k2\pi\)

a/ \(y=2\left(\frac{\sqrt{3}}{2}sinx-\frac{1}{2}cosx\right)+5=2sin\left(x-\frac{\pi}{6}\right)+5\)

Do \(-1\le sin\left(x-\frac{\pi}{6}\right)\le1\Rightarrow3\le y\le7\)

b/ \(y=2cos\left(x+\frac{\pi}{6}\right)cos\left(-\frac{\pi}{6}\right)=\sqrt{3}cos\left(x+\frac{\pi}{6}\right)\)

Do \(-1\le cos\left(x+\frac{\pi}{6}\right)\le1\Rightarrow-\sqrt{3}\le y\le\sqrt{3}\)

c/ \(y=2\left(\frac{1}{2}sinx+\frac{\sqrt{3}}{2}cosx\right)+12=2sin\left(x+\frac{\pi}{3}\right)+12\)

Do \(-1\le sin\left(x+\frac{\pi}{3}\right)\le1\Rightarrow10\le y\le14\)

f(

Đặt \(sinx+cosx=\sqrt{2}sin\left(x+\dfrac{\pi}{4}\right)=t\Rightarrow t\in\left[-\sqrt{2};\sqrt{2}\right]\)

\(t^2=1+2sinx.cosx\Rightarrow sinx.cosx=\dfrac{t^2-1}{2}\)

\(\Rightarrow y=t+\dfrac{t^2-1}{2}=\dfrac{1}{2}t^2+t-\dfrac{1}{2}\)

Xét hàm \(y=f\left(t\right)=\dfrac{1}{2}t^2+t-\dfrac{1}{2}\) trên \(\left[-\sqrt{2};\sqrt{2}\right]\)

\(-\dfrac{b}{2a}=-1\in\left[-\sqrt{2};\sqrt{2}\right]\)

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

\(\Rightarrow y_{min}=-1\) ; \(y_{max}=\dfrac{1+2\sqrt{2}}{2}\)

\(y=sinx.cosx\left(sin^2x-cos^2x\right)=\frac{1}{2}sin2x.\left(-cos2x\right)=-\frac{1}{4}sin4x\)

Do \(-1\le sin4x\le1\Rightarrow-\frac{1}{4}\le y\le\frac{1}{4}\)

\(y_{min}=-\frac{1}{4}\) khi \(sin4x=1\)

\(y_{max}=\frac{1}{4}\) khi \(sin4x=-1\)

\(y=sin\left(x+\dfrac{\pi}{3}\right)-sinx\)

\(=\dfrac{1}{2}sinx+\dfrac{\sqrt{3}}{2}cosx-sinx\)

\(=\dfrac{\sqrt{3}}{2}cosx-\dfrac{1}{2}sinx\)

\(=cos\left(x+\dfrac{\pi}{6}\right)\in\left[-1;1\right]\)

\(\Rightarrow\left\{{}\begin{matrix}y_{mịn}=-1\Leftrightarrow x=\dfrac{5\pi}{6}+k2\pi\\y_{max}=1\Leftrightarrow x=-\dfrac{\pi}{6}+k2\pi\end{matrix}\right.\)

a: ĐKXĐ; 1-sin x>=0

=>sin x<=1(luôn đúng)

b: ĐKXĐ: 1-cosx>=0

=>cosx<=1(luôn đúng)

c: ĐKXĐ: 1-cos²x>=0

=>cos²x<=1

=>-1<=cosx<=1(luôn đúng)