x2+2y2=1 求2x+3y2的最值

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x2+2y2=1求2x+3y2的最值x2+2y2=1求2x+3y2的最值x2+2y2=1求2x+3y2的最值y^2=(1-x^2)/22x+3y2=2x+3/2-3/2y^2=-3/2(x-2/3)^

x2+2y2=1 求2x+3y2的最值
x2+2y2=1 求2x+3y2的最值

x2+2y2=1 求2x+3y2的最值
y^2=(1-x^2)/2
2x+3y2=2x+3/2-3/2y^2
=-3/2(x-2/3)^2+2/3+3/2
x∈[-1,1]
∴2x+3y^2最大值为x=2/3时,=13/6
最小值为x=-1时,=-2

由x²+2y²=1，得 y²=(1-x²)/2≥0 -1≤x≤1
则 2x+3y²=2x+3(1-x²)/2=-3(x-2/3)²/2+13/6
又 -1≤x≤1，得最大值13/6(x=2/3),最小值-2(x=-1)

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