Cho a,b,c\(\ge0\),thoa man a+b+c=4.Chung minh rang:
\(\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\ge4\)
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Cho 3 so duong a,b,c thoa man dieu kien : a+b+c=1. Chung minh rang
\(\sqrt{4a+1}+\sqrt{4b+1}+\sqrt{4c+1}< 5\)
Áp dụng BĐT Bunhia:
\(\sqrt{4a+1}+\sqrt{4b+1}+\sqrt{4c+1}\le\sqrt{\left(1+1+1\right)\left(4a+1+4b+1+4c+1\right)}\)
\(\Rightarrow\sqrt{4a+1}+\sqrt{4b+1}+\sqrt{4c+1}\le\sqrt{3.\left(4\left(a+b+c\right)+3\right)}=\sqrt{21}< \sqrt{25}=5\)
Vậy \(\sqrt{4a+1}+\sqrt{4b+1}+\sqrt{4c+1}< 5\)
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cho a, b duong thoan man a+b=c. Chung minh rang \(\sqrt[4]{a^3}+\sqrt[4]{b^3}>\sqrt[4]{c^3}\)
cho a,b,c la ba so thuc duong thoa man dieu kien a+b+c=1
chung minh rang P=\(\sqrt{\frac{ab}{c+ab}}+\sqrt{\frac{bc}{a+bc}}+\sqrt{\frac{ca}{b+ca}}\le\frac{3}{2}\)
lấy bút xóa mà xóa hết là khỏe
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cho a,b,c la do dai 3 canh cua mot tam giac thoa man dieu kien \(\sqrt{a+b-c}+\sqrt{b+c-a}+\sqrt{c+a-b}=\sqrt{a}+\sqrt{b}+\sqrt{c}\)
chung minh a,b,c la 3 canh cua mot tam giac deu
cho a,b,c thoa man \(a\times\sqrt{1-b^2}+b\times\sqrt{1-c^2}+c\times\sqrt{1-a^2}=\dfrac{3}{2}\)
chung minh \(a^2+b^2+c^2=\dfrac{3}{2}\)
cho ca so a,b,c duong thoa man ab+bc+ca =1 chung minh \(P=\frac{2a}{\sqrt{1+a^2}}+\frac{b}{\sqrt{1+b^2}}+\frac{c}{\sqrt{1+c^2}}\le\frac{1}{4}\)
cho so thuc a,b,c voi a ,b duong va c\(\ne\)0 thoa man
\(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=0\)
1/chung minh c<0 , a+c>0 va b+c >0
2/chung minh \(\sqrt{a+b}=\sqrt{a+c}+\sqrt{b+c}\)
cho cac so a,b,c duong thoa man ab+bc+ca=1 chung minh : \(p=\frac{2a}{\sqrt{1+a^2}}+\frac{b}{\sqrt{1+b^2}}+\frac{c}{\sqrt{1+c^2}}\)
1)Rut gon bieu thuc:P=\(\sqrt{4+\sqrt{10+2\sqrt{5}}}+\sqrt{4-\sqrt{10+2\sqrt{5}}}\)
2) Cho 3 so duong a,b,c thoa man dieu kien:a+b+c=\(\sqrt{ab}+\sqrt{bc}+\sqrt{ca}\)
Chung minh rang:a=b=c