Bài 1 Tìm giá trị lớn nhất , giá trị nhỏ nhất ( nếu có ) của hàm số sau :
6 , \(y=cos^2x+2sinx+2\)
7 , \(y=sin^4-2cos^2x+1\)
8 , \(y=\frac{1+4cos^2x}{3}\)
9 , \(y=\sqrt{1+sin2x}\)
10 , \(y=3-4sin^2x.cos^2x\)
12 , \(y=8+\frac{1}{2}sinx.cosx\)
13 \(y=\frac{1+4sin^2x}{3}\)
15 , \(y=\sqrt{1-sin\left(x^2\right)}-1\)
16 , \(y=2cos\left(x+\frac{\pi}{3}\right)+3\)
17 , \(y=\sqrt{1-cosx}\)
19 , \(y=\sqrt{5-2sin^2xcos^2x}\)
21 , \(y=2sin^2x-cos2x\)
23 , \(y=\frac{2}{1+tan^2x}\)
24 , \(y=\frac{1}{cosx+1}\)
6.
\(y=1-sin^2x+2sinx+2=-sin^2x+2sinx+3\)
\(y=-\left(sinx-1\right)^2+4\le4\)
\(y_{max}=4\) khi \(sinx=1\)
\(y=\left(sinx+1\right)\left(3-sinx\right)\ge0\)
\(y_{min}=0\) khi \(sinx=-1\)
7.
\(y=sin^4x-2\left(1-sin^2x\right)+1=sin^4x+2sin^2x-1\)
Do \(0\le sin^2x\le1\Rightarrow-1\le y\le2\)
\(y_{min}=-1\) khi \(sin^2x=0\)
\(y_{max}=2\) khi \(sin^2x=1\)
8.
\(y=\frac{1}{3}+\frac{4}{3}cos^2x\)
Do \(0\le cos^2x\le1\Rightarrow\frac{1}{3}\le y\le\frac{5}{3}\)
\(y_{min}=\frac{1}{3}\) khi \(cos^2x=0\)
\(y_{max}=\frac{5}{3}\) khi \(cos^2x=1\)
9.
\(-1\le sin2x\le1\Rightarrow0\le1+sin2x\le2\)
\(\Rightarrow0\le y\le\sqrt{2}\)
\(y_{min}=0\) khi \(sin2x=-1\)
\(y_{max}=\sqrt{2}\) khi \(sin2x=1\)
10.
\(y=3-\left(2sinx.cosx\right)^2=3-sin^22x\)
Do \(0\le sin^22x\le1\Rightarrow2\le y\le3\)
\(y_{min}=2\) khi \(sin^22x=1\)
\(y_{max}=3\) khi \(sin2x=0\)
12.
\(y=8+\frac{1}{4}\left(2sinx.cosx\right)=8+\frac{1}{4}sin2x\)
Do \(-1\le sin2x\le1\Rightarrow\frac{31}{4}\le y\le\frac{33}{4}\)
\(y_{min}=\frac{31}{4}\) khi \(sin2x=-1\)
\(y_{max}=\frac{33}{4}\) khi \(sin2x=1\)
13.
Về bản chất giống hệt câu 13, chỉ cần thay chữ sin bằng chữ cos
15.
\(-1\le sin\left(x^2\right)\le1\Rightarrow-1\le y\le\sqrt{2}-1\)
\(y_{min}=-1\) khi \(sin\left(x^2\right)=1\)
\(y_{max}=\sqrt{2}-1\) khi \(sin\left(x^2\right)=-1\)
16.
\(-1\le cos\left(x+\frac{\pi}{3}\right)\le1\Rightarrow1\le y\le5\)
\(y_{min}=1\) khi \(cos\left(x+\frac{\pi}{3}\right)=-1\)
\(y_{max}=5\) khi \(cos\left(x+\frac{\pi}{3}\right)=1\)
17.
\(-1\le cosx\le1\Rightarrow0\le y\le\sqrt{2}\)
\(y_{min}=0\) khi \(cosx=1\)
\(y_{max}=\sqrt{2}\) khi \(cosx=-1\)
19.
\(y=\sqrt{5-\frac{1}{2}\left(2sinxcosx\right)^2}=\sqrt{5-\frac{1}{2}sin^22x}\)
\(0\le sin^22x\le1\Rightarrow\frac{3\sqrt{2}}{2}\le y\le\sqrt{5}\)
\(y_{min}=\frac{3\sqrt{2}}{2}\) khi \(sin^22x=1\)
\(y_{max}=\sqrt{5}\) khi \(sin^22x=0\)
21.
\(y=2sin^2x-\left(1-2sin^2x\right)=4sin^2x-1\)
\(0\le sin^2x\le1\Rightarrow-1\le y\le3\)
\(y_{min}=-1\) khi \(sin^2x=0\)
\(y_{max}=3\) khi \(sin^2x=1\)
23.
\(tan^2x\ge0\Rightarrow y\le2\)
\(y_{max}=2\) khi \(tanx=0\)
\(y_{min}\) không tồn tại
24.
\(-1\le cosx\le1\Rightarrow0< 1+cosx\le2\)
\(\Rightarrow y\ge\frac{1}{2}\)
\(y_{min}=\frac{1}{2}\) khi \(cosx=1\)
\(y_{max}\) ko tồn tại
