bui hong diep

1) Cmr: $\forall \alpha$ ta có: $\sqrt{17}\leq \sqrt{cos^{2}\alpha -2cos\alpha +3}+\sqrt{cos^{2}\alpha +4cos\alpha +6}\leq \sqrt{2}+\sqrt{11}$

2) Cho $ a,b,c,d$ tm: $a^{2}+b^{2}=c^{2}+d^{2}=5.$. Cmr:

a) $\sqrt{5-a-2b}+\sqrt{5-c-2d}+\sqrt{5-ac-bd}\leq \frac{3\sqrt{30}}{2}$

b) $(5-a-2b)\left ( 5-c-2d \right )\left (5-ac-bd \right )\leq \left ( \frac{15}{2} \right )^{2}$

3) Cmr: 

$\sqrt{2x^{2}-2x+1}+\sqrt{2x^{2}-(\sqrt{3}-1)x+1}+\sqrt{2x^{2}+(\sqrt{3}+1)x+1}\geq 3$

tìm các số thực a,b,c tm: |ax+by+cz| + |bx+cy+az| + |cx+ay+bz| = |x| + |y| + |z| 

Cho x, y, z $\epsilon \mathbb{R}$ đôi 1$\neq$ nhau tm:

$\left ( y-z \right )\sqrt[3]{1-x^{3}} + \left ( z-x \right )\sqrt[3]{1-y^{3}}+\left ( x-y \right )\sqrt[3]{1-z^{3}}=0$

CMR: $\left ( 1-x^{3} \right )\left ( 1-y^{3} \right )\left ( 1-z^{3} \right )=\left ( 1-xyz \right )^{3}$

$\left ( x+y+z \right )\left ( \frac{1}{x} +\frac{1}{y}+\frac{1}{z}\right )\geq 9$

cho $0^{\circ}< \alpha < 90^{\circ}$

1) Tìm GTLN của: A = $sin^{2}\alpha .cos\alpha ^{4}$

2) Tìm GTNN của:A = $\frac{1}{sin\alpha .cos\alpha }$

                             B = $\frac{1}{cos^{2}\alpha }+\frac{1}{sin^{2}\alpha }$

                             C = $\frac{2}{cos^{2}\alpha }+3cot^{2}\alpha$