用均值不等式求y=2x^2+1/x+1(x>-1)的最值

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用均值不等式求y=2x^2+1/x+1(x>-1)的最值

用均值不等式求y=2x^2+1/x+1(x>-1)的最值
用均值不等式求y=2x^2+1/x+1(x>-1)的最值

用均值不等式求y=2x^2+1/x+1(x>-1)的最值
y=(2x²+1)/(x+1)
=[2(x+1-1)² +1]/(x+1)
=[2(x+1)² -4(x+1)+3]/(x+1)
=2(x+1)+3/(x+1) -4
≥2√[2(x+1)•3/(x+1)]-4
=2√6 -4
当且仅当 2(x+1)=3/(x+1),即x=√6/4 -1时,
y有最小值为2√6 -4