MT 【331】两元非齐次不等式
若正实数$x,y$满足$x^3+y^3=(4x-5y)y$ 则 $y$ 的最大值为____
解答:$x^3+y^3+y^2=4(x-y)y\le x^2$,故$y^3+y^2=x^2-x^3=\dfrac{x(2-2x)x}{2}\le\dfrac{4}{27}$,故由$f(t)=t^3+t^2$的单调性$y\le \dfrac{1}{3}$
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若正实数$x,y$满足$x^3+y^3=(4x-5y)y$ 则 $y$ 的最大值为____
解答:$x^3+y^3+y^2=4(x-y)y\le x^2$,故$y^3+y^2=x^2-x^3=\dfrac{x(2-2x)x}{2}\le\dfrac{4}{27}$,故由$f(t)=t^3+t^2$的单调性$y\le \dfrac{1}{3}$