Question

[Hoang Thao Bich Tram]

Tìm GTNN bằng cách dùng bất đẳng thức|a|+|b| lớn hơn hoặc bằng |a+b|

Tìm GTNN

C= 3.|x+2|+|3x+1|

Answer 1

[Bùi_Hoàng_Yến]

\(|a+b|\ge0\)\(\Rightarrow GTNN|a+b|=0\)

\(|a|\ge0;|b|\ge0\Rightarrow a=0;b=0\)

\(C=3|x+2|+|3x+1|\)

\(\hept{\begin{cases}|x+2|\ge0\Rightarrow3|x+2|\ge0\\|3x+1|\ge0\end{cases}}\)

\(\Rightarrow\hept{\begin{cases}GTNN3|x+2|=0\\GTNN|3x+1|=0\end{cases}}\Rightarrow C=0\)

\(\hept{\begin{cases}3|x+2|=0\Rightarrow|x+2|=0\Rightarrow x+2=0\Rightarrow x=-2\\|3x+1|=0\Rightarrow3x+1=0\Rightarrow3x=-1\Rightarrow x=\frac{-1}{3}\end{cases}}\)

\(\Rightarrow x\)không thể có 2 giá trị.\(\Rightarrow\orbr{\begin{cases}3|x+2|=0\\|3x+1|=0\end{cases}}\Rightarrow\orbr{\begin{cases}x=-2\\x=\frac{-1}{3}\end{cases}}\)

Xét \(x=-2\)và\(x=\frac{-1}{3}\):

\(x=-2\Rightarrow3|x+2|=0\Rightarrow C=|3x+1|\)

\(C1=|3x+1|\)

   \(=|3.\left(-2\right)+1|\)

   \(=|\left(-6\right)+1|\)

   \(=|-5|\)

   \(=5\)

\(x=\frac{-1}{3}\Rightarrow|3x+1|=0\Rightarrow C=3|x+2|\)

\(C2=3|x+2|\)

   \(=3|\frac{-1}{3}+2|\)

   \(=3|\frac{-1+6}{3}|\)

   \(=3|\frac{5}{3}|\)

   \(=\frac{3.5}{3}\)

   \(=5\)

\(C1=C2=5\)

\(\Rightarrow GTNNC=5\)