已知a,b,c为正实数,且a^2+b^2+c^2=1 求证 a(1-a^2)<=(2√3)⼀9

2024-12-15 00:14:49
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证明:a,b,c为正实数,令a^2=b^2=c^2,则:a^2=b^2=c^2=1/3
a(1-a^2)=√(1/3)*(1-1/3)=(√3)/3 *2/3=(2√3)/9
而当a^2=b^2=c^2时a(1-a^2)最大,故:a(1-a^2)<=(2√3)/9