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Wildan Bagus W

Bilangan kompleks

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Diberikan $a$ dan $b$ bilangan kompleks yang memenuhi

\begin{align*} \begin{cases} a^2 + b^2 &=7 \\ a^3 + b^3 &= 10 \end{cases} \end{align*}

Tentukan nilai maksimum dari $a+b$. 

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We know that $(a-b)^2\geq0$. Then \begin{align*}a^2+b^2&\geq2ab\\ab&\leq\frac{a^2+b^2}{2}\\ab&\leq\frac{7}{2}.\end{align*} From the equation given, \begin{align*}(a^2+b^2)(a+b)&=a^3+b^3+ab(a+b)\\7(a+b)&=10+ab(a+b)\\7(a+b)&\leq10+\frac{7}{2}(a+b)\\\frac{7}{2}(a+b)&\leq10\\a+b&\leq\frac{20}{7}.\end{align*} So, the maximum value of $a+b$ is $\boxed{\frac{20}{7}}$.

Edited by Gethux

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