Tìm giá trị lớn nhất của E=-8x^2-3y^2-26x+6y+100

\(E=-8x^2-3y^2-26x+6y+100\)

\(E=\left(-8x^2-26x\right)+\left(-3y^2+6y\right)+100\)

\(E=-8.\left(x^2+3,25x\right)-3.\left(y^2-2y\right)+100\)

\(E=-8.\left(x^2+1,625x+1,625x+\dfrac{169}{64}-\dfrac{169}{64}\right)-3.\left(y^2-y-y-1+1\right)+100\)

\(E=-8.\left[\left(x^2+1,625x\right)+\left(1,625x+\dfrac{169}{64}\right)-\dfrac{169}{64}\right]-3.\left[\left(y^2-y\right).\left(y-1\right)+1\right]+100\)

\(E=-8.\left[x.\left(x+1,625\right)+1,625.\left(x+1,625\right)-\dfrac{169}{64}\right]-3.\left[y.\left(y-1\right).\left(y-1\right)+1\right]+100\)

\(E=-8.\left[\left(x+1,625\right)^2-\dfrac{169}{64}\right]-3.\left[\left(y-1\right)^2+1\right]+100\)

\(E=-8.\left(x+1,625\right)^2+\dfrac{169}{8}-3.\left(y-1\right)^2-3.\left(y-1\right)^2-3+100\)

\(E=-8.\left(x+1,625\right)^2-3.\left(y-1\right)^2+\dfrac{945}{8}\)

\(E=-\left[8.\left(x+1,625\right)^2+3.\left(y-1\right)^2\right]+\dfrac{945}{8}\)

Với mọi giá trị của \(x;y\in R\) ta có:

\(8.\left(x+1,625\right)^2\ge0;3.\left(y-1\right)^2\ge0\)

\(\Rightarrow8.\left(x+1,625\right)^2+3.\left(y-1\right)^2\ge0\)

\(\Rightarrow-\left(8.\left(x+1,625\right)^2+3.\left(y-1\right)^2\right)\le0\)

\(-\left(8.\left(x+1,625\right)^2+3.\left(y-1\right)^2\right)+\dfrac{945}{8}\le\dfrac{945}{8}\)

Hay \(E\le\dfrac{945}{8}\) với mọi giá trị của \(x;y\in R\)

Để \(E=\dfrac{945}{8}\) thì \(-\left(8.\left(x+1,625\right)^2+3.\left(y-1\right)^2\right)+\dfrac{945}{8}=\dfrac{945}{8}\)

\(\Rightarrow-\left(8.\left(x+1,625\right)^2+3.\left(y-1\right)^2\right)=0\)

\(\Rightarrow8.\left(x+1,625\right)^2+3.\left(y-1\right)^2=0\)

\(\Rightarrow\left\{{}\begin{matrix}8.\left(x+1,625\right)^2=0\\3.\left(y-1\right)^2=0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}\left(x+1,625\right)^2=0\\\left(y-1\right)^2=0\end{matrix}\right.\)

\(\Rightarrow\left\{{}\begin{matrix}x+1,625=0\\y-1=0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}x=-1,625\\y=1\end{matrix}\right.\)

Vậy giá trị nhỏ nhất của biểu thức E là \(\dfrac{945}{8}\) đạt được khi và chỉ khi \(x=-1,625;y=1\)

Chúc bạn học tốt nha!!!