Question

ex 10 b, c dau coroană! dau 100 de pcte pls

Answer 1

b)

$\it m_a\geq m_g \\ \\ \dfrac{1+a}{2}\geq\sqrt{1\cdot a} \Rightarrow 1+a\geq2\sqrt a\ \ \ \ (1) \\ \\ \dfrac{1+b}{2}\geq\sqrt{1\cdot b} \Rightarrow 1+b\geq2\sqrt b\ \ \ \ (2)\\ \\ \dfrac{1+c}{2}\geq\sqrt{1\cdot c} \Rightarrow 1+c\geq2\sqrt c\ \ \ \ \ (3)$

$\it (1),(2),(3) \Rightarrow (1+a)(1+b)(1+c)\geq8\sqrt{abc} =8\cdot\sqrt1=8$

c)

$\it m_g\leq m_a\\ \\ a,b>0 \Rightarrow 1+a, 1+b >1 \Rightarrow 1+a\ne1,\ \ 1+b\ne1\\ \\ \\ \sqrt{1\cdot(1+a)} <\dfrac{1+1+a}{2} \Rightarrow \sqrt{1+a}<\dfrac{2+a}{2}\ \ \ \ (1)\\ \\ \\ \sqrt{1\cdot(1+b)} <\dfrac{1+1+b}{2} \Rightarrow \sqrt{1+b}<\dfrac{2+b}{2}\ \ \ \ \ (2)\\ \\ \\ (1),(2) \Rightarrow \sqrt{1+a}+\sqrt{1+b} < \dfrac{2+a+2+b}{2}=\dfrac{4+(a+b)}{2} =\dfrac{4+4}{2}=2$