Bài 2

5 C

Bài 3

1 D

6 C

Còn lại ol r nhé

2) 5. C

3) 2. D

6. C

Còn lại ok nha

2.

\(cosx+cos3x=1+\sqrt{2}sin\left(2x+\dfrac{\pi}{4}\right)\)

\(\Leftrightarrow2cos2x.cosx=1+cos2x+sin2x\)

\(\Leftrightarrow2cos2x.cosx=2cos^2x+2sinx.cosx\)

\(\Leftrightarrow cosx\left(cos2x-cosx-sinx\right)=0\)

\(\Leftrightarrow cosx\left(cos^2x-sin^2x-cosx-sinx\right)=0\)

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

\(\Leftrightarrow cosx.\sqrt{2}sin\left(x+\dfrac{\pi}{4}\right).\left[\sqrt{2}cos\left(x+\dfrac{\pi}{4}\right)-1\right]=0\)

\(\Leftrightarrow\left[{}\begin{matrix}cosx=0\\sin\left(x+\dfrac{\pi}{4}\right)=0\\cos\left(x+\dfrac{\pi}{4}\right)=\dfrac{1}{\sqrt{2}}\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{2}+k\pi\\x=-\dfrac{\pi}{4}+k\pi\\x+\dfrac{\pi}{4}=\pm\dfrac{\pi}{4}+k2\pi\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\pi}{2}+k\pi\\x=-\dfrac{\pi}{4}+k\pi\\x=k2\pi\end{matrix}\right.\)

Mình cảm ơn

2b)

Áp dụng BĐT bunhiacopxki có:

\(\left(1+1\right)\left(x^4+y^4\right)\ge\left(x^2+y^2\right)^2\)

\(\left(1+1\right)\left(x^2+y^2\right)\ge\left(x+y\right)^2\)\(\Leftrightarrow x^2+y^2\ge\dfrac{\left(x+y\right)^2}{2}\)

\(\Rightarrow2\left(x^4+y^4\right)\ge\dfrac{\left(x+y\right)^4}{4}\Leftrightarrow x^4+y^4\ge\dfrac{1}{8}.\left(x+y\right)^4\)

Dấu "=" xảy ra khi x=y

3)

Áp dụng bđt Holder có:

\(\left(x^3+y^3+z^3\right)\left(1+1+1\right)\left(1+1+1\right)\ge\left(x+y+z\right)^3\)

\(\Leftrightarrow x^3+y^3+z^3\ge\dfrac{1}{9}\left(x+y+z\right)^3\)

Dấu "=" xảy ra khi x=y=z

3)(Nếu không dùng Holder)

Với x,y,z >0, ta có bđt sau:\(2x^3+2y^3+2z^3\ge xy\left(x+y\right)+yz\left(y+z\right)+xz\left(x+z\right)\) (1)

Thật vậy (1)\(\Leftrightarrow\left(x+y\right)\left(x^2-xy+y^2\right)-xy\left(x+y\right)+\left(y+z\right)\left(y^2-yz+z^2\right)-yz\left(y+z\right)+\left(z+x\right)\left(z^2-zx+x^2\right)-zx\left(x+z\right)\ge0\)

\(\Leftrightarrow\left(x+y\right)\left(x-y\right)^2+\left(y+z\right)\left(y-z\right)^2+\left(z+x\right)\left(z-x\right)^2\ge0\) (lđ)

Áp dụng AM-GM có:

\(x^3+y^3+z^3\ge3xyz\)

\(\Leftrightarrow\dfrac{2\left(x^3+y^3+z^3\right)}{3}\ge2xyz\) (2)

Từ (1) và (2), cộng vế với vế \(\Rightarrow\dfrac{8}{3}\left(x^3+y^3+z^3\right)\ge xy\left(x+y\right)+yz\left(x+z\right)+xz\left(x+z\right)+2xyz\)

\(\Leftrightarrow\dfrac{8}{3}\left(x^3+y^3+z^3\right)\ge\left(x+y\right)\left(y+z\right)\left(z+x\right)\)

\(\Leftrightarrow8\left(x^3+y^3+z^3\right)\ge3\left(x+y\right)\left(y+z\right)\left(x+z\right)\)

\(\Leftrightarrow9\left(x^3+y^3+z^3\right)\ge x^3+y^3+z^3+3\left(x+y\right)\left(y+z\right)\left(z+x\right)=\left(x+y+z\right)^3\)

\(\Rightarrow x^3+y^3+z^3\ge\dfrac{1}{9}\left(x+y+z\right)^3\) (đpcm)

\(3,\\ a,ĐK:x\ge-5\\ PT\Leftrightarrow2\sqrt{x+5}-2\sqrt{x+5}+3\sqrt{x+5}=12\\ \Leftrightarrow\sqrt{x+5}=4\Leftrightarrow x+5=16\Leftrightarrow x=11\left(tm\right)\\ b,ĐK:x\in R\\ PT\Leftrightarrow\left|x-5\right|=6\Leftrightarrow\left[{}\begin{matrix}x-5=6\\5-x=6\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=11\\x=-1\end{matrix}\right.\)

lần sau bạn chụp thẳng ra nha,đùng chụp ngang

e thay dấu = cho tất cả phsố trog bài 3 rồi tìm x , khi tìm x thì coi dấu của bài r nói x lớn hoặc nhỏ hơn số đó là đc

\(\dfrac{9^{15}.8^{11}}{3^{29}.16^8}=\dfrac{\left(3^2\right)^{15}.\left(2^3\right)^{11}}{3^{29}.\left(2^4\right)^8}=\dfrac{3^{30}.2^{33}}{3^{29}.2^{32}}\)

Ta lấy vễ trên chia vế dưới

\(=3.2=6\)

\(\dfrac{2^{11}.9^3}{3^5.16^2}=\dfrac{2^{11}.\left(3^2\right)^3}{3^5.\left(2^4\right)^2}=\dfrac{2^{11}.3^6}{3^5.2^8}\)

Ta lấy vế trên chia vế dưới

\(=2^3.3=24\)

\(\dfrac{9^{15}.8^{11}}{3^{29}.16^8}=\dfrac{\left(3^2\right)^{15}.\left(2^3\right)^{11}}{3^{29}.\left(2^4\right)^8}=\dfrac{3^{30}.2^{33}}{3^{29}.3^{32}}=3.2=6\)
\(\dfrac{2^{11}.9^3}{3^5.16^2}=\dfrac{2^{11}.\left(3^2\right)^3}{3^5.\left(2^4\right)^2}=\dfrac{2^{11}.3^6}{3^5.2^8}=2^3.3=8.3=24\)

3:

c: Xét ΔCAM có KI//AM

nên KI/AM=CI/CM

Xét ΔCMB có HI//MB

nên HI/MB=CI/CM

=>KI/AM=HI/MB

=>KI=HI

=>I là trung điểm của HK

III

1 It's colder today than yesterday

2 It takes 4 hours to travel by car and fives hours by train

3 We were busier at work today than everyday

4 Jane's sister cooks worse than her

5 Nobody in this team can play football as well as Tom

IV

1 D
2 A