Question

[Trần Long]

 

Answer 1

[Toru]

Ta thấy: \(\left\{{}\begin{matrix}\left(3x-2\right)^{2022}\ge0;\forall x\\\left(5y+4\right)^{2024}\ge0;\forall y\end{matrix}\right.\)

\(\Rightarrow\left(3x-2\right)^{2022}+\left(5y+4\right)^{2024}\ge0;\forall x,y\)

\(\Rightarrow\left(3x-2\right)^{2022}+\left(5y+4\right)^{2024}+2023\ge2023;\forall x,y\)

\(\Rightarrow C\ge2023;\forall x,y\)

Dấu \("="\) xảy ra khi: \(\left\{{}\begin{matrix}3x-2=0\\5y+4=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=\dfrac{2}{3}\\y=-\dfrac{4}{5}\end{matrix}\right.\)

Vậy \(C_{min}=2023\) tại \(x=\dfrac{2}{3};y=-\dfrac{4}{5}\).