Toán [toán 9] bất đẳng thức

R

riverflowsinyou1

$a+b$ \geq $c$ \geq $b+1$ \Leftrightarrow $a$ \geq $1$

Mà $b$ \geq $a$ \Leftrightarrow $b$\geq1

\Rightarrow $(a-1)(b-1)$ \geq 0 \Leftrightarrow $ab+1$\geq$a+b$\geqc \Leftrightarrow $ab$\geq$c-1$

Có $Q$\geq$\dfrac{2ab+abc}{(a+1)(b+1)(c-1)}=\dfrac{ab(c+2)}{(ab+a+b+1)(c+1)}$

\Rightarrow $Q$ \geq $\dfrac{ab(c+2)}{2(ab+1)(c+1)}=\dfrac{c+2}{2(1+ \dfrac{1}{ab} )(c+1)}$

\Rightarrow $Q$ \geq $\dfrac{c+2}{2(1+\dfrac{1}{c-1})(c+1)}=\dfrac{(c+2)(c-1)}{2c(c+1)}$

\Leftrightarrow $Q$ \geq $\dfrac{c^2+c-2}{2(c^2+c)}=\dfrac{1}{2}-\dfrac{1}{c^2+c}$

\Leftrightarrow $Q$ \geq $\dfrac{1}{2}-\dfrac{1}{3^2+3}=\dfrac{5}{12}$
 
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