Question

Tìm giá trị lớn nhất của biểu thức:

\(A=4x-x^2+3\)

\(B=x-x^2\)

\(C=2x-2x^2-5\)

Answer 1

+) ta có : \(A=4x-x^2+3=-\left(x^2-4x+4\right)+7\)

\(=-\left(x-2\right)^2+7\le7\) \(\Rightarrow A_{max}=7\) khi \(x=2\)

+) ta có : \(B=x-x^2=-\left(x^2-x+\dfrac{1}{4}\right)+\dfrac{1}{4}\)

\(=-\left(x-\dfrac{1}{2}\right)^2+\dfrac{1}{4}\le\dfrac{1}{4}\) \(\Rightarrow B_{max}=\dfrac{1}{4}\) khi \(x=\dfrac{1}{2}\)

+) ta có : \(C=2x-2x^2-5=-2\left(x^2-x+\dfrac{1}{4}\right)-\dfrac{9}{2}\)

\(=-2\left(x-\dfrac{1}{2}\right)^2-\dfrac{9}{2}\le\dfrac{-9}{2}\) \(\Rightarrow C_{max}=\dfrac{-9}{2}\) khi \(x=\dfrac{1}{2}\)