Tìm giá trị lớn nhất, giá trị nhỏ nhất của (x-2)^2-1

a)\(\left(x-2\right)^2-1\)

Dễ thấy:\(\left(x-2\right)^2\ge0\forall x\)

\(\Rightarrow\left(x-2\right)^2-1\ge-1\forall x\)

Đẳng thức xảy ra khi \(x=2\)

b)\(\left(x^2-9\right)^2+\left|y-2\right|+10\)

Dễ thấy: \(\left\{{}\begin{matrix}\left(x^2-9\right)^2\ge0\\\left|y-2\right|\ge0\end{matrix}\right.\)

\(\Rightarrow\left(x^2-9\right)^2+\left|y-2\right|\ge0\)

\(\Rightarrow\left(x^2-9\right)^2+\left|y-2\right|+10\ge10\)

Đẳng thức xảy ra khi \(\left\{{}\begin{matrix}x^2-9=0\\y-2=0\end{matrix}\right.\)\(\left\{{}\begin{matrix}x=\pm3\\y=2\end{matrix}\right.\)

c)\(\dfrac{3}{\left(x-2\right)^2+5}\)

Dễ thấy:

\(\left(x-2\right)^2\ge0\forall x\Rightarrow\left(x-2\right)^2+5\ge5\)

\(\Rightarrow\dfrac{1}{\left(x-2\right)^2+5}\le\dfrac{1}{5}\Rightarrow\dfrac{3}{\left(x-2\right)^2+5}\le\dfrac{3}{5}\)

Đẳng thức xảy ra khi \(x-2=0\Rightarrow x=2\)

d)\(-10-\left(x-30\right)^2-\left|y-5\right|\)

Dễ thấy: \(\left\{{}\begin{matrix}\left(x-30\right)^2\ge0\\\left|y-5\right|\ge0\end{matrix}\right.\)\(\Rightarrow\left\{{}\begin{matrix}-\left(x-30\right)^2\le0\\-\left|y-5\right|\le0\end{matrix}\right.\)

\(\Rightarrow-\left(x-30\right)^2-\left|y-5\right|\le0\)

\(\Rightarrow10-\left(x-30\right)^2-\left|y-5\right|\le10\)

Đẳng thức xảy ra khi \(\Rightarrow\left\{{}\begin{matrix}-\left(x-30\right)^2=0\\-\left|y-5\right|=0\end{matrix}\right.\)\(\Rightarrow\left\{{}\begin{matrix}x=30\\y=5\end{matrix}\right.\)