Răspuns:
Explicație pas cu pas:
a)
[tex]a=\frac{12}{\sqrt{6}} + \sqrt{(1-\sqrt{6})^2} + |-1| =[/tex]
[tex]=\frac{12*\sqrt{6} }{\sqrt{6}*\sqrt{6} } + |(1-\sqrt{6}| +1 =[/tex]
[tex]=\frac{12*\sqrt{6} }{6 } + \sqrt{6}-1 + 1 =[/tex]
[tex]=2*\sqrt{6} + \sqrt{6} =[/tex]
[tex]=3*\sqrt{6}[/tex]
[tex]b = |2-\sqrt{12}|+2*\sqrt{(-1+\sqrt{3}-3*\sqrt{6} )^2} =[/tex]
[tex]= \sqrt{12}-2 +2*|-1+\sqrt{3}-3*\sqrt{6}| =[/tex]
[tex]= \sqrt{4*3}-2 +2*(1-\sqrt{3}+3*\sqrt{6}) =[/tex]
[tex]= \sqrt{4}*\sqrt{3} -2 +2-2*\sqrt{3}+6*\sqrt{6} =[/tex]
[tex]= 2*\sqrt{3}-2*\sqrt{3}+6*\sqrt{6} =[/tex]
[tex]= 6*\sqrt{6}[/tex]
m=min{a;b} = min(3√6 ; 6√6) = 3√6 = a
M=max{a;b} = max(3√6 ; 6√6) = 6√6 = b
b)
[tex]ma = \frac{a+b}{2} = \frac{3*\sqrt{3}+6*\sqrt{3} }{2} = \frac{\sqrt{3}*(3+6) }{2} = \frac{9*\sqrt{3}}{2} =4,5*\sqrt{3}[/tex]
[tex]mg = \sqrt{a*b} = \sqrt{3*\sqrt{3} *6*\sqrt{3} } = \sqrt{3*6*\sqrt{3}*\sqrt{3} }=[/tex]
[tex]= \sqrt{3*6*3 } = \sqrt{3*3}*\sqrt{6} = 3*\sqrt{6}[/tex]
Asadar:
m = mg = [tex]3*\sqrt{6}[/tex] <[tex]4,5*\sqrt{6}[/tex] = ma < [tex]6*\sqrt{6}[/tex] = M
Deci
m ≤ mg ≤ma ≤M.
