Question

Cho a²+4b²+9c²=1. Tìm min và max của a+b+c

Answer 1

Đặt \(P=a+b+c\)

\(P^2=\left(a+b+c\right)^2=\left(1.a+\dfrac{1}{2}.2b+\dfrac{1}{3}.3c\right)^2\le\left(1^2+\left(\dfrac{1}{2}\right)^2+\left(\dfrac{1}{3}\right)^2\right)\left(a^2+4b^2+9c^2\right)\)

\(\Rightarrow P^2\le\dfrac{49}{36}\left(a^2+4b^2+9c^2\right)=\dfrac{49}{36}\)

\(\Rightarrow-\dfrac{7}{6}\le P\le\dfrac{7}{6}\)

\(P_{min}=-\dfrac{7}{6}\) khi \(\left(a;b;c\right)=\left(-\dfrac{6}{7};-\dfrac{3}{14};-\dfrac{2}{21}\right)\)

\(P_{max}=\dfrac{7}{6}\) khi \(\left(a;b;c\right)=\left(\dfrac{6}{7};\dfrac{3}{14};\dfrac{2}{21}\right)\)